CMU 15-896 Noncooperative games 4: Stackelberg games Teacher: - - PowerPoint PPT Presentation

CMU 15-896

Noncooperative games 4: Stackelberg games

Teacher: Ariel Procaccia

15896 Spring 2016: Lecture 20

A curious game

• Playing up is a dominant

strategy for row player

• So column player would

play left

• Therefore,

is the

• nly Nash equilibrium
• utcome

2

1,1 3,0 0,0 2,1

15896 Spring 2016: Lecture 20

Commitment is good

• Suppose the game is played

as follows:

• Row player commits to

playing a row

• Column player observes the

commitment and chooses column

• Row player can commit to

playing down!

3

1,1 3,0 0,0 2,1

15896 Spring 2016: Lecture 20

Commitment to mixed strategy

• By committing to a

mixed strategy, row player can guarantee a reward of 2.5

• Called a Stackelberg

(mixed) strategy

4

1 .49 1,1

3,0

.51 0,0

2,1

15896 Spring 2016: Lecture 20

Computing Stackelberg

• Theorem [Conitzer and Sandholm 2006]:

In 2-player normal form games, an optimal Stackelberg strategy can be found in poly time

• Theorem [ditto]: the problem is NP-hard

when the number of players is  3

5

15896 Spring 2016: Lecture 20

Tractability: 2 players

• For each pure follower strategy , we compute via

the LP below a strategy for the leader such that

• Playing is a best response for the follower
• Under this constraint, is optimal
• Choose

∗ that maximizes leader value

6

max ∑ ,

∈

s.t. ∀

∈ ,

∀ ∈ , ∈ 0,1 ∑ , ∑ ,

• ∈

∈

∑ 1

∈

15896 Spring 2016: Lecture 20

Application: security

7

• Airport security:

deployed at LAX

• Federal Air Marshals
• Coast Guard
• Idea:
• Defender commits to

mixed strategy

• Attacker observes and

best responds

15896 Spring 2016: Lecture 20

security games

• Set of targets
• Set of

security resources available to the defender (leader)

• Set of schedules
• Resource

can be assigned to one of the schedules in

• Attacker chooses one target

to attack

8

resources targets

15896 Spring 2016: Lecture 20

security games

• For each target , there are four

numbers:

• , and
• Let
• be the

vector of coverage probabilities

• The utilities to the

defender/attacker under c if target is attacked are

• 9

resources targets

15896 Spring 2016: Lecture 20

10

This is a 2-player Stackelberg game. Can we compute an optimal strategy for the defender in polynomial time?

15896 Spring 2016: Lecture 20

Solving security games

• Consider the case of

, i.e., resources are assigned to individual targets, i.e., schedules have size 1

• Nevertheless, number of leader strategies is

exponential

• Theorem [Korzhyk et al. 2010]: Optimal

leader strategy can be computed in poly time

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15896 Spring 2016: Lecture 20

A compact LP

• LP formulation

similar to previous

• ne
• Advantage:

logarithmic in #leader strategies

• Problem: do

probabilities correspond to strategy?

12

max ∗, s.t. ∀ ∈ Ω, ∀ ∈ , 0 , 1 ∀ ∈ ,

• , 1

∈:∈

∀ ∈ Ω, , 1

∈

∀ ∈ , , ∗,

15896 Spring 2016: Lecture 20

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• 0.7

0.2 0.1 0.3 0.7

• 0.7 0.2 0.1
• 0.3 0.7
• 1
• 1
• 1

1

• 1

1

• 1

1

15896 Spring 2016: Lecture 20

Fixing the probabilities

• Theorem [Birkhoff-von Neumann]: Consider an matrix

with real numbers ∈ 0,1, such that for each , ∑ 1

• ,

and for each , ∑ 1

• is kinda doubly stochastic). Then

there exist matrices , … , and weights , … , such that:

1.

∑ 1

• 2.

∑

• 3.

For each , is kinda doubly stochastic and its elements are in 0,1

• The probabilities , satisfy theorem’s conditions
• By 3, each is a deterministic strategy
• By 1, we get a mixed strategy
• By 2, gives right probs

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15896 Spring 2016: Lecture 20

Generalizing?

• What about schedules of

size 2?

• Air Marshals domain has

such schedules:

• utgoing+incoming flight

(bipartite graph)

• Previous apporoach fails
• Theorem [Korzhyk et al.

2010]: problem is NP-hard

15

0.5 0.5 0.5 0.5

15896 Spring 2016: Lecture 20

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15896 Spring 2016: Lecture 20

Criticisms

• Problematic assumptions:

1.

The attacker exactly observes the defender’s mixed strategy

2.

The defender knows the attacker’s utility function

3.

The attacker behaves in a perfectly rational way

• We will focus on relaxing assumption #1

17

15896 Spring 2016: Lecture 20

Limited surveillance

• Let us compare two worlds:

1.

Status quo: The defender optimizes against an attacker with unlimited observations (i.e., complete knowledge of the defender’s strategy), but the attacker actually has only

• bservations

2.

Ideal: The defender optimizes against an attacker with

• bservations, and,

miraculously, the attacker indeed has exactly

• bservations

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15896 Spring 2016: Lecture 20

Limited surveillance

• Theorem [Blum et al. 2014]: Assume that

utilities are normalized to be in . For any , there is a zero-sum security game such that the difference between worlds and is

• Lemma: If
• , there exists
• such that:

1.

∀, ||/2

2.

Each ∈ is in exactly members of

3.

If ⊂ and then ⋃

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2

15896 Spring 2016: Lecture 20

Proof of theorem

• resources, each can defend any

targets,

• ,
• targets
• For any target , zero-sum utilities with
• and
• Poll: The optimal strategy (in the status

quo world) defends each target with probability roughly…?

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15896 Spring 2016: Lecture 20

Proof of theorem

• Next we define a much better strategy against an

attacker with observations

• subset of targets 1, … ,
• ⊆
• Define , … as in the lemma
• Pure strategy covers ; this is valid because

/2 (by property 1)

• Let ∗ be the uniform distribution over , … ,
• By property 2, ∗ covers each target in with

probability ½

• By property 3, observations from ∗ would show some

target in never being covered; that target is attacked ∎

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15896 Spring 2016: Lecture 20

Limited surveillance

• Theorem [Blum et al. 2014]: For any zero-

sum security game with targets, resources, and a set of schedules with max coverage , and for any

• bservations, the difference between the

two worlds is at most

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