[PPT] - Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, PowerPoint Presentation

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Cops and Robbers: The Cost of Drunkenness

Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki March 24, 2014

This research has been co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) Research Funding Program: THALIS UOA (MIS 375891). Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 1 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

The Cost of Drunkenness

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 2 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

The Cost of Drunkenness

In “classic” CR game both cop and robber are adversarial

The cop moves to capture the robber as fast as possible The robber moves to avoid capture for as long as possible

What if the robber is drunk?, i.e., he performs a random walk. The adversarial cop should be able to capture the drunk robber no later than the adversarial one. We will try to quantify this intuition Two papers:

Some remarks on cops and drunk robbers (Kehagias + Pralat, TCS 2012) Cops and invisible robbers: The cost of drunkenness (Kehagias + Mitsche + Pralat, TCS 2013)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 2 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

The Cost of Drunkenness

In “classic” CR game both cop and robber are adversarial

The cop moves to capture the robber as fast as possible The robber moves to avoid capture for as long as possible

What if the robber is drunk?, i.e., he performs a random walk. The adversarial cop should be able to capture the drunk robber no later than the adversarial one. We will try to quantify this intuition Two papers:

Some remarks on cops and drunk robbers (Kehagias + Pralat, TCS 2012) Cops and invisible robbers: The cost of drunkenness (Kehagias + Mitsche + Pralat, TCS 2013)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 2 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Outline

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

A pair (σC, σR) determines a time of capture T(sc, sR) as follows: sC ∗ sR = x0y0x1...xtyt... T(sC, sR) = min{t : xt = yt−1 or yt = xt} (with min ∅ = ∞).

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

A pair (σC, σR) determines a time of capture T(sc, sR) as follows: sC ∗ sR = x0y0x1...xtyt... T(sC, sR) = min{t : xt = yt−1 or yt = xt} (with min ∅ = ∞).

The payoff to R is T (sC, sR)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

A pair (σC, σR) determines a time of capture T(sc, sR) as follows: sC ∗ sR = x0y0x1...xtyt... T(sC, sR) = min{t : xt = yt−1 or yt = xt} (with min ∅ = ∞).

The payoff to R is T (sC, sR) The payoff to C is −T (sC, sR)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

A pair (σC, σR) determines a time of capture T(sc, sR) as follows: sC ∗ sR = x0y0x1...xtyt... T(sC, sR) = min{t : xt = yt−1 or yt = xt} (with min ∅ = ∞).

The payoff to R is T (sC, sR) The payoff to C is −T (sC, sR)

Hence the AV-CR is a two-player, zero-sum, perfect information game.

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Visible Drunk Robber

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33

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t=0

Payoff is expected capture time E(T|sC) = 1 + E(

∞

∑

t=1

1(xt / ∈ {yt−1, yt}) which C wants to minimize. This is an infinite horizon, undiscounted MDP problem. Theorem There exists a memoryless strategy σ∗

C s.t.

E(T|σ∗

C) = inf sC E(T|sC).

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Visible Drunk Robber

The stochastic process (xt, yt)∞

t=0 is a Markov Decision

Process (MDP) with control variable (xt)∞

t=0

Payoff is expected capture time E(T|sC) = 1 + E(

∞

∑

t=1

1(xt / ∈ {yt−1, yt}) which C wants to minimize. This is an infinite horizon, undiscounted MDP problem. Theorem There exists a memoryless strategy σ∗

C s.t.

E(T|σ∗

C) = inf sC E(T|sC).

Then dct(G) = E(T|σ∗

C) is the drunk capture time.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Cost of Drunkenness

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Cost of Drunkenness

Now we define the cost of drunkenness: F(G) =

ct(G) dct(G)

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is a video; something is wrong with it ... Here is another video

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Strategy now is a function sC or sR s.t. Pr(xt+1|x0y0...xtyt) = sC (x0y0...xtyt) Pr(yt+1|x0y0...ytxt+1) = sR (x0y0...ytxt+1) Payoff is expected capture time E(T|sC, sR) AI-CR is a two-player, zero-sum game but without perfect information. Theorem There exist strategies s∗

C, s∗ R such that

E(T|s∗

C, s∗ R) = sup sR

inf

sC E(T|sC, sR) = inf sC sup sR

E(T|sC, sR).

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Invisible Adversarial Robber

Strategy now is a function sC or sR s.t. Pr(xt+1|x0y0...xtyt) = sC (x0y0...xtyt) Pr(yt+1|x0y0...ytxt+1) = sR (x0y0...ytxt+1) Payoff is expected capture time E(T|sC, sR) AI-CR is a two-player, zero-sum game but without perfect information. Theorem There exist strategies s∗

C, s∗ R such that

E(T|s∗

C, s∗ R) = sup sR

inf

sC E(T|sC, sR) = inf sC sup sR

E(T|sC, sR). Then cti(G) = E(T|σ∗

C, σ∗ R) is the invisible capture

time.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Invisible Adversarial Robber

Strategy now is a function sC or sR s.t. Pr(xt+1|x0y0...xtyt) = sC (x0y0...xtyt) Pr(yt+1|x0y0...ytxt+1) = sR (x0y0...ytxt+1) Payoff is expected capture time E(T|sC, sR) AI-CR is a two-player, zero-sum game but without perfect information. Theorem There exist strategies s∗

C, s∗ R such that

E(T|s∗

C, s∗ R) = sup sR

inf

sC E(T|sC, sR) = inf sC sup sR

E(T|sC, sR). Then cti(G) = E(T|σ∗

C, σ∗ R) is the invisible capture

time.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Invisible Drunk Robber

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33

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t=0

Payoff is expected capture time E(T|sC) = 1 + E(

∞

∑

t=1

1(xt / ∈ {yt−1, yt}) which C wants to minimize. This is an infinite horizon, undiscounted POMDP problem. Theorem There exists a memoryless strategy σ∗

C s.t.

E(T|σ∗

C) = inf sC E(T|sC).

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Invisible Drunk Robber

The stochastic process (xt, yt)∞

t=0 is a Partially Observable

MDP (POMDP) with control variable (xt)∞

t=0

Payoff is expected capture time E(T|sC) = 1 + E(

∞

∑

t=1

1(xt / ∈ {yt−1, yt}) which C wants to minimize. This is an infinite horizon, undiscounted POMDP problem. Theorem There exists a memoryless strategy σ∗

C s.t.

E(T|σ∗

C) = inf sC E(T|sC).

Then dcti(G) = E(T|σ∗

C) is the invisible drunk capture

time.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Invisible Cost of Drunkenness

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Invisible Cost of Drunkenness

Define the invisible cost of drunkenness: Fi(G) =

cti(G) dcti(G)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Invisible Cost of Drunkenness

Define the invisible cost of drunkenness: Fi(G) =

cti(G) dcti(G)

Question: What values can Fi(G) take?

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Additional Results

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Additional Results

Invisibility does not increase the cop number

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Additional Results

Invisibility does not increase the cop number General lower bound on dcti(G): dcti(G) ≥

δ(n−K) 7e∆·c(G)

SLIDE 86

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Additional Results

Invisibility does not increase the cop number General lower bound on dcti(G): dcti(G) ≥

δ(n−K) 7e∆·c(G)

General upper bound on cti(G): cti(G) ≤ ( ˆ T + D)(∆ + 1) ˆ

T·n

For AV-CR and DV-CR, optimal strategies exist for any graph (not only cop-win) We have computed cti(G), dcti(G), Fi(G) for specific graph families Graph Fi(G) Path 2 + o(1) Cycle 2 + o(1) Tree Θ(log n) Grid ≤ O(n3/25

√n)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is a video

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 14 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is a video; something is wrong with it ...

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 14 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: The Cost of Visibility

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33

SLIDE 90

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: The Cost of Visibility

COV: H(G) = cti(G)

ct(G)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33

SLIDE 91

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: The Cost of Visibility

COV: H(G) = cti(G)

ct(G)

Drunk COV: Hd(G) = dcti(G)

dct(G)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33

SLIDE 92

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: The Cost of Visibility

COV: H(G) = cti(G)

ct(G)

Drunk COV: Hd(G) = dcti(G)

dct(G)

The Role of Visibility in Pursuit / Evasion Games, Kehagias + Mitsche + Pralat, arXiv:1402.6136

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33

SLIDE 93

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 94

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 95

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 96

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 97

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 98

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 99

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 100

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

The lazy robber is also a FSA (how many states?)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 101

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

The lazy robber is also a FSA (how many states?) In general, consider FSA with m states (m ∈ {1, 2, ..., n2)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 102

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

The lazy robber is also a FSA (how many states?) In general, consider FSA with m states (m ∈ {1, 2, ..., n2) There must exist an optimal (wrt capture time) robber R(m) of m states

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 103

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

The lazy robber is also a FSA (how many states?) In general, consider FSA with m states (m ∈ {1, 2, ..., n2) There must exist an optimal (wrt capture time) robber R(m) of m states We expect that capture time increases with m

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 104

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

The lazy robber is also a FSA (how many states?) In general, consider FSA with m states (m ∈ {1, 2, ..., n2) There must exist an optimal (wrt capture time) robber R(m) of m states We expect that capture time increases with m Of course optimality also depends on the type of cop

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 105

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

The lazy robber is also a FSA (how many states?) In general, consider FSA with m states (m ∈ {1, 2, ..., n2) There must exist an optimal (wrt capture time) robber R(m) of m states We expect that capture time increases with m Of course optimality also depends on the type of cop Generally: study game of k-state robber against m-state robber

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 106

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Extra Topic: Cop / Robber Automata

Simple example, just the beginning of an idea Suppose CR is played on a path Pn Drunk robber plays like a stochastic FSA with a single state Adversarial robber also plays like a stochastic FSA

The FSA has one state for every (x, y) position (so n2 states total) For every state there is a (trivial) next state transition probability.

The lazy robber is also a FSA (how many states?) In general, consider FSA with m states (m ∈ {1, 2, ..., n2) There must exist an optimal (wrt capture time) robber R(m) of m states We expect that capture time increases with m Of course optimality also depends on the type of cop Generally: study game of k-state robber against m-state robber

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33

SLIDE 107

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Existence of Optimal Strategies

CR-DV: memoryless optimal strategies exist (MDP literature)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 17 / 33

SLIDE 108

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Existence of Optimal Strategies

CR-DV: memoryless optimal strategies exist (MDP literature) CR-DI: optimal strategies exist (KMP)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 17 / 33

SLIDE 109

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Existence of Optimal Strategies

CR-DV: memoryless optimal strategies exist (MDP literature) CR-DI: optimal strategies exist (KMP) CR-AI: optimal strategies exist (Gurevich, KMP)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 17 / 33

SLIDE 110

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Existence of Optimal Strategies

CR-DV: memoryless optimal strategies exist (MDP literature) CR-DI: optimal strategies exist (KMP) CR-AI: optimal strategies exist (Gurevich, KMP) CR-AV: memoryless optimal strategies exist (Hahn,MacGillivray,Bonato)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 17 / 33

SLIDE 111

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Existence of Optimal Strategies

CR-DV: memoryless optimal strategies exist (MDP literature) CR-DI: optimal strategies exist (KMP) CR-AI: optimal strategies exist (Gurevich, KMP) CR-AV: memoryless optimal strategies exist (Hahn,MacGillivray,Bonato) ???

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 17 / 33

SLIDE 112

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Existence of Optimal Strategies

CR-DV: memoryless optimal strategies exist (MDP literature) CR-DI: optimal strategies exist (KMP) CR-AI: optimal strategies exist (Gurevich, KMP) CR-AV: memoryless optimal strategies exist (Hahn,MacGillivray,Bonato) ???

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 17 / 33

SLIDE 113

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Optimal Strategies for CR-AV

Proved in (Hahn+MacGillivray, 2006), for more general games in (Bonato+MacGillivray, 2013)

SLIDE 120

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

A2′: No advantage in remembering how a position was reached

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 19 / 33

SLIDE 121

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

SLIDE 130

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Zermelo and K¨

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

In 1913 Zermelo claimed that: if a player can win, he can win in bounded time

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

SLIDE 131

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Zermelo and K¨

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

In 1913 Zermelo claimed that: if a player can win, he can win in bounded time (bound independent of opponent’s strategy)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

SLIDE 132

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Zermelo and K¨

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

In 1913 Zermelo claimed that: if a player can win, he can win in bounded time (bound independent of opponent’s strategy) He argued that the player can win w/o repeating any positions

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

SLIDE 133

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Zermelo and K¨

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

In 1913 Zermelo claimed that: if a player can win, he can win in bounded time (bound independent of opponent’s strategy) He argued that the player can win w/o repeating any positions In 1927 K¨

nig argued that Zermelo’s claim is true, but his

proof is false

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

SLIDE 134

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Zermelo and K¨

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

In 1913 Zermelo claimed that: if a player can win, he can win in bounded time (bound independent of opponent’s strategy) He argued that the player can win w/o repeating any positions In 1927 K¨

nig argued that Zermelo’s claim is true, but his

proof is false and provided an alternative proof

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

SLIDE 135

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Zermelo and K¨

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

In 1913 Zermelo claimed that: if a player can win, he can win in bounded time (bound independent of opponent’s strategy) He argued that the player can win w/o repeating any positions In 1927 K¨

nig argued that Zermelo’s claim is true, but his

proof is false and provided an alternative proof Zermelo accepted K¨

nig’s arguments

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

SLIDE 136

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Zermelo and K¨

nig

Is A1 true? This question was discussed in the context of chess in 1913-1927 by Zermelo and D. K¨

nig

In 1913 Zermelo claimed that: if a player can win, he can win in bounded time (bound independent of opponent’s strategy) He argued that the player can win w/o repeating any positions In 1927 K¨

nig argued that Zermelo’s claim is true, but his

proof is false and provided an alternative proof Zermelo accepted K¨

nig’s arguments

This story is told in (Schwalbe + Walker, 2001)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 20 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that:

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 138

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 139

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves This follows from game theoretic results:

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 140

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves This follows from game theoretic results:

every finite game of perfect information has memoryless

ptimal strategies (A2)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 141

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves This follows from game theoretic results:

every finite game of perfect information has memoryless

ptimal strategies (A2)

From A2 we easily get no repeated positions

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 142

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves This follows from game theoretic results:

every finite game of perfect information has memoryless

ptimal strategies (A2)

From A2 we easily get no repeated positions

Using bounded time, we can approximate the full game (of potentially infinite duration)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 143

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves This follows from game theoretic results:

every finite game of perfect information has memoryless

ptimal strategies (A2)

From A2 we easily get no repeated positions

Using bounded time, we can approximate the full game (of potentially infinite duration) by finite duration games

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 144

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves This follows from game theoretic results:

every finite game of perfect information has memoryless

ptimal strategies (A2)

From A2 we easily get no repeated positions

Using bounded time, we can approximate the full game (of potentially infinite duration) by finite duration games Instead of using this route, let’s exploit an alternative, interesting connection

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 21 / 33

SLIDE 145

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Game Theory Note that: bounded time → no repeated moves This follows from game theoretic results:

every finite game of perfect information has memoryless

ptimal strategies (A2)

A reachability game is played by two players on a digraph G =

V , E
Move: slide token from node to node (along an edge)

The i-th player slides the token iff it is on a node v ∈ V i

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 22 / 33

SLIDE 149

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Reachability Games

A reachability game is played by two players on a digraph G =

V , E
Move: slide token from node to node (along an edge)

The i-th player slides the token iff it is on a node v ∈ V i Player 0 wins iff the token goes into a node u ∈ F

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 22 / 33

SLIDE 150

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Reachability Games

A reachability game is played by two players on a digraph G =

V , E
Move: slide token from node to node (along an edge)

The i-th player slides the token iff it is on a node v ∈ V i Player 0 wins iff the token goes into a node u ∈ F Player 1 wins otherwise

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 22 / 33

SLIDE 151

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Reachability Games

A reachability game is played by two players on a digraph G =

V , E
Move: slide token from node to node (along an edge)

The i-th player slides the token iff it is on a node v ∈ V i Player 0 wins iff the token goes into a node u ∈ F Player 1 wins otherwise The game is fully described by

V 0, V 1, E, F
Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki

Cops and Robbers: The Cost of Drunkenness March 24, 2014 22 / 33

SLIDE 152

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Reachability Games

A reachability game is played by two players on a digraph G =

V , E
Move: slide token from node to node (along an edge)

The i-th player slides the token iff it is on a node v ∈ V i Player 0 wins iff the token goes into a node u ∈ F Player 1 wins otherwise The game is fully described by

V 0, V 1, E, F
Theorem

Let

V 0, V 1, E, F
be a reachability game on the digraph

D =

V , E
. Then V can be partitioned into two sets W 0 and

W 1 such that (for i ∈ {0, 1}) player i has a memoryless strategy σi which is winning whenever the game starts in u ∈ W i.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 22 / 33

SLIDE 153

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Reachability Games

A reachability game is played by two players on a digraph G =

V , E
Move: slide token from node to node (along an edge)

The i-th player slides the token iff it is on a node v ∈ V i Player 0 wins iff the token goes into a node u ∈ F Player 1 wins otherwise The game is fully described by

V 0, V 1, E, F
Theorem

Let

V 0, V 1, E, F
be a reachability game on the digraph

D =

V , E
. Then V can be partitioned into two sets W 0 and

W 1 such that (for i ∈ {0, 1}) player i has a memoryless strategy σi which is winning whenever the game starts in u ∈ W i.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 22 / 33

SLIDE 154

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

CR is a Reachability Game

Construct the move digraph (ala Hahn+MacGillivray)

SLIDE 159

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

CR is a Reachability Game

Construct the move digraph (ala Hahn+MacGillivray)

The CR move digraph MG has a node for every position (x, y, p), where

x is the cop location y is the robber location p is the player to move

Edges are added accordingly

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 23 / 33

SLIDE 160

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

CR is a Reachability Game

Construct the move digraph (ala Hahn+MacGillivray)

The CR move digraph MG has a node for every position (x, y, p), where

x is the cop location y is the robber location p is the player to move

Edges are added accordingly

This MG can be used to play any modified CR game (with prespecified initial player positions and starting player)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 23 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

CR is a Reachability Game

Construct the move digraph (ala Hahn+MacGillivray)

The CR move digraph MG has a node for every position (x, y, p), where

x is the cop location y is the robber location p is the player to move

Edges are added accordingly

This MG can be used to play any modified CR game (with prespecified initial player positions and starting player) To include the “classic” CR we must add nodes

1 (∅, ∅, C), 2 (x, ∅, C) (x ∈ |V |),

and corresponding edges

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 23 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

2 V 0 contains the nodes (x, y, C)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

2 V 0 contains the nodes (x, y, C) 3 V 1 contains the nodes (x, y, R)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

SLIDE 166

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

2 V 0 contains the nodes (x, y, C) 3 V 1 contains the nodes (x, y, R) 4 F (the cop’s target set) contains the nodes (x, x, p)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

SLIDE 167

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

2 V 0 contains the nodes (x, y, C) 3 V 1 contains the nodes (x, y, R) 4 F (the cop’s target set) contains the nodes (x, x, p)

This RG subsumes both classic and modified CR games

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

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Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

2 V 0 contains the nodes (x, y, C) 3 V 1 contains the nodes (x, y, R) 4 F (the cop’s target set) contains the nodes (x, x, p)

This RG subsumes both classic and modified CR games The classic CR game is the RG started at (∅, ∅, C)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

SLIDE 169

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

2 V 0 contains the nodes (x, y, C) 3 V 1 contains the nodes (x, y, R) 4 F (the cop’s target set) contains the nodes (x, x, p)

This RG subsumes both classic and modified CR games The classic CR game is the RG started at (∅, ∅, C) The graph is cop-win if (∅, ∅, C) ∈ W0; otherwise it is robber-win

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

SLIDE 170

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Consider the RG

V 0, V 1, E, F
, where

1 MG =

V , E
is the “expanded” game digraph.

2 V 0 contains the nodes (x, y, C) 3 V 1 contains the nodes (x, y, R) 4 F (the cop’s target set) contains the nodes (x, x, p)

This RG subsumes both classic and modified CR games The classic CR game is the RG started at (∅, ∅, C) The graph is cop-win if (∅, ∅, C) ∈ W0; otherwise it is robber-win

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 24 / 33

SLIDE 171

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

If the graph is cop-win, the cop has a memoryless winning strategy when starting from (∅, ∅, C)

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 25 / 33

SLIDE 172

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

If the graph is cop-win, the cop has a memoryless winning strategy when starting from (∅, ∅, C) Still do not know if a time optimal memoryless cop strategy exists.

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 25 / 33

SLIDE 173

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

If the graph is cop-win, the cop has a memoryless winning strategy when starting from (∅, ∅, C) Still do not know if a time optimal memoryless cop strategy exists. But we can leverage the fact that game duration is bounded

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 25 / 33

SLIDE 174

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

If the graph is cop-win, the cop has a memoryless winning strategy when starting from (∅, ∅, C) Still do not know if a time optimal memoryless cop strategy exists. But we can leverage the fact that game duration is bounded Similar things for robber-win graphs

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 25 / 33

SLIDE 175

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

If the graph is cop-win, the cop has a memoryless winning strategy when starting from (∅, ∅, C) Still do not know if a time optimal memoryless cop strategy exists. But we can leverage the fact that game duration is bounded Similar things for robber-win graphs Similar things for graphs with cop number c(G) ≥ K

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 25 / 33

SLIDE 176

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

If the graph is cop-win, the cop has a memoryless winning strategy when starting from (∅, ∅, C) Still do not know if a time optimal memoryless cop strategy exists. But we can leverage the fact that game duration is bounded Similar things for robber-win graphs Similar things for graphs with cop number c(G) ≥ K And we can (should?) define cop number a little differently

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 25 / 33

SLIDE 177

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

If the graph is cop-win, the cop has a memoryless winning strategy when starting from (∅, ∅, C) Still do not know if a time optimal memoryless cop strategy exists. But we can leverage the fact that game duration is bounded Similar things for robber-win graphs Similar things for graphs with cop number c(G) ≥ K And we can (should?) define cop number a little differently

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 25 / 33

SLIDE 178

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is the video again

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 26 / 33

SLIDE 179

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is the video again What is wrong with the video?

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 26 / 33

SLIDE 180

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is the video again What is wrong with the video? Cops and robber move “concurrently”!

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 26 / 33

SLIDE 181

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is the video again What is wrong with the video? Cops and robber move “concurrently”! For applications CR game must have “concurrent” moves

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 26 / 33

SLIDE 182

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Here is the video again What is wrong with the video? Cops and robber move “concurrently”! For applications CR game must have “concurrent” moves

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 26 / 33

SLIDE 183

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

M. Dille, “Search and Pursuit with Unmanned Aerial

Vehicles in Road”, Ph.D. Thesis, 2013 Robotics Institute, Carnegie Mellon University

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 28 / 33

SLIDE 188

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Helicopter Search (really: Unmanned Aerial Vehicle (UAV) Search)

M. Dille, “Search and Pursuit with Unmanned Aerial

Vehicles in Road”, Ph.D. Thesis, 2013 Robotics Institute, Carnegie Mellon University Problem: With one or more UAVs locate an evasive target within an urban-like road network

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 28 / 33

SLIDE 189

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Helicopter Search (really: Unmanned Aerial Vehicle (UAV) Search)

M. Dille, “Search and Pursuit with Unmanned Aerial

Vehicles in Road”, Ph.D. Thesis, 2013 Robotics Institute, Carnegie Mellon University Problem: With one or more UAVs locate an evasive target within an urban-like road network

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 28 / 33

SLIDE 190

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

An Unmanned Aerial Vehicle

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 29 / 33

SLIDE 191

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

An Environment

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 30 / 33

SLIDE 192

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Search Algorithm

One-step leaf optimization Search subtrees in increasing order by search number Pre-position sweepers waiting on guards Optimize assignment, sequence, and direction of leaf subtrees

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 31 / 33

SLIDE 193

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

UAV Search Examples

Simple environment Real environment: Capture Real environment: Escape

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 32 / 33

SLIDE 194

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Rescue in Contaminated Environment

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 33 / 33

SLIDE 195

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Rescue in Contaminated Environment An indoor environment

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 33 / 33

SLIDE 196

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Rescue in Contaminated Environment An indoor environment The black rectangle is an exit

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 33 / 33

SLIDE 197

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Rescue in Contaminated Environment An indoor environment The black rectangle is an exit The smiley face is a human

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 33 / 33

SLIDE 198

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Rescue in Contaminated Environment An indoor environment The black rectangle is an exit The smiley face is a human The little cloud is a leak of toxic gas

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 33 / 33

SLIDE 199

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Invisible Cost

Application: Rescue in Contaminated Environment An indoor environment The black rectangle is an exit The smiley face is a human The little cloud is a leak of toxic gas The Problem: At time t = 0 the human is situated at node u1 and a toxic leak appears at node u2. The human wants to reach the exit node u0

Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 33 / 33