Improved Equilibria via Public Service Advertising

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Improved Equilibria via Public Service Advertising

• Maria-Florina Balcan

Microsoft Research
Joint with Avrim Blum and Yishay Mansour
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Good equilibria, Bad equilibria
Many 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.

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Good equilibria, Bad equilibria
Many games have both good and bad equilibria.
Fair cost-sharing.

• 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.

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Good equilibria, Bad equilibria
Many games have both good and bad equilibria.
Fair cost-sharing.

• 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.

Good equilibrium all use edge of cost 1.
(paying 1/n each)
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Good equilibria, Bad equilibria
Many games have both good and bad equilibria.
Fair cost-sharing.

• 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.

Good equilibrium all use edge of cost 1.
(paying 1/n each)
Bad equilibrium all use edge of cost n-?.
(paying 1- ²/n each)
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Good equilibria, Bad equilibria
Many games have both good and bad equilibria.
Fair cost-sharing.

• 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.

v

• Bad eq. result of natural dynamics
• players entering one at time
• minimizing regret

Subway/shared van
cars
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Good equilibria, Bad equilibria
Standard motivation for PoS
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.
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.
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Good equilibria, Bad equilibria
What if only some ? fraction will pay
attention?
Ride public transit
Fundamental Questions

• Can the authority guide behavior to a good state?
  

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

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Main Model
0. n players initially playing some arbitrary
equilibrium.

1. Authority launches advertising, proposing joint
   action sad.

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

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

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

Notes

• Only consider potential games.

• Focus on social cost

(Except we use makespan for load balancing.)
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Main Results
Cost sharing
(PoS log(n), PoA n)

• 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.

(PoS 1, PoA 1)
Load Balancing

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

Party Affiliation
(PoS 1, PoA ?(n2))

• Threshold behavior for ? gt ½, can get ratio
  O(1), but for ? lt ½, ratio stays ?(n2). (assume
  degrees ?(log n)).

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Fair Cost Sharing
Cost sharing
(PoS log(n), PoA n)
If only a constant fraction ? of the players
follow the advice, then we get within O(1/?) of
the PoS.
Note this is best you can hope for. E.g., k
2?n.
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Fair Cost Sharing
Cost sharing
(PoS log(n), PoA n)
If only a constant fraction ? of the players
follow the advice, then we get within O(1/?) of
the PoS.
Proof Idea
- Advertiser proposes OPT (any apx also works)
- In any NE a non-receptive player i, cant
improve by switching to his path PiOPT in OPT.
- Moreover, this option is guaranteed to be at
least as good as if other NR players didnt exist.
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Fair Cost Sharing
Cost sharing
(PoS log(n), PoA n)
If only a constant fraction ? of the players
follow the advice, then we get within O(1/?) of
the PoS.
Proof Idea
- Advertiser proposes OPT (any apx also works)
- In any NE a non-receptive player i, cant
improve by switching to his path PiOPT in OPT.
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Fair Cost Sharing
Cost sharing
(PoS log(n), PoA n)
If only a constant fraction ? of the players
follow the advice, then we get within O(1/?) of
the PoS.
Proof Idea
- Advertiser proposes OPT (any apx also works)
- In any NE a non-receptive player i, cant
improve by switching to his path PiOPT in OPT.
- Calculate total cost of these guaranteed
options.
- Rearrange sum...
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Fair Cost Sharing
Cost sharing
(PoS log(n), PoA n)
If only a constant fraction ? of the players
follow the advice, then we get within O(1/?) of
the PoS.
Proof Idea
- Advertiser proposes OPT (any apx also works)
- In any NE a non-receptive player i, cant
improve by switching to his path PiOPT in OPT.
- 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 Sharing
Cost sharing
(PoS log(n), PoA n)
If only a constant fraction ? of the players
follow the advice, then we get within O(1/?) of
the PoS.
Proof Idea
- Finally, in last phase, std potential argument
shows behavior cannot get worse by more than an
additional log(n) factor.
(End of phase 3)
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Cost Sharing, Extension
Cost sharing
linear delays
- Still get same guarantee, but proof is trickier
- Problem cant argue as if remaining NR players
didnt exist since they add to delays
Proof Idea
- Define shadow game pure linear latency fns.
Offset defined by equilib at end of phase 2.
- This has good PoA.
users on e at end of phase 2
- Behavior at end of phase 2 is equilib for this
game too.
- Show
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Party 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 games
OPT is an equilibrium so PoS 1.
But even for consensus, PoA ?(n2)
all edges labeled plus
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Party affiliation games
Party Affiliation
(PoS 1, PoA ?(n2))
- Threshold behavior for ? gt ½, can get ratio
O(1), but for ? lt ½, ratio stays ?(n2). (assume
degrees ?(log n)).
Proof Idea
(lower bound)
- Same example as for consensus PoA, but sparser
across cut. Players locked into place.
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Party affiliation games
Party Affiliation
(PoS 1, PoA ?(n2))
- Threshold behavior for ? gt ½, can get ratio
O(1), but for ? lt ½, ratio stays ?(n2). (assume
degrees ?(log n)).
Proof Idea
(upper bound)
- Split nodes into those incurring low-cost vs
those incurring high-cost under OPT.
- Advertising strategy follow OPT.
- Show that low-cost will switch to behavior in
OPT. For high-cost, dont care.
- Cost only improves in final best-response
process.
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Conclusions and Open Questions
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?