Prologue

• http://www.youtube.com/watch?v=XP9cfQx2OZY

Truels and N-uelsAn Analysis

Background: Truels

• Truels in the Movies: – The Good, The Bad, and The Ugly– Reservoir Dogs– Pulp Fiction– Pirates of the Caribbean

• Animal Behavior: Three Fierce Animal Rivals living in Proximity, but without significant aggression?

• Real Showdowns: ABC, NBC, and CBS competition for late night audience.

Past Studies and Our Focus

Our first interest in the truel came from the paper: “The Truel”

by D. Marc Kilgour and Steven J. Brams.The paper discussed:• Sequential (fixed order): The players fire one at a

time in a fixed, repeating sequence, such as A,B,C,A,B,C,A...

• Sequential (random order): The first player to fire, and each subsequent player, is chosen at random among the survivors.

• Simultaneous: All surviving players fire simultaneously in every round.

An Example and Previous Research

Let’s consider one situation in Truels and Nuels :

- each player is a perfect shot

- each shoots in a sequence.

An Example and Previous Research

Taking Turns From 1 Player’s Perspective

1st Shooter: A

A shoots B

C can then shoot at A.

An Example and Previous Research

However, if A shoots into the air,

B’s response should be to eliminate his only threat, C.

Realism: Sequential vs Simultaneous

Sequential Please wait your turn to be shot.Some Rules agreed upon for further exploration:• Each player prefers an outcome in which

he/she survives to one in which he/she does not survive.

• Players continue to fire until only one survives.• Simultaneous: All surviving players fire

simultaneously in every round.

Further Exploration

What if they are not perfect

shots ? What if they have more than one

shot ? What if they shoot at the same

time ? Who will a player choose to shoot

? How ? Is conditional probability a viable

tool of analysis ? What kinds of mathematical tools

will we need ? How will we generalize for use in

similar scenarios ?

Truel Simulation

• Conditional Probability becomes exponentially more complex.

Application Developed to run Truel Simulations Advantages:

Configurable Settings-Number of Players-Strategy Used-Number of Full Rounds to Run

Acc1 Acc2 Acc3 Str1 Str2 Str3 Wins1 Wins2 Wins3 None

0.91 0.9 0.89 Best Best Best 9 81 9146 764

0.91 0.9 0.89 Worst Best Best 9 9102 88 801

0.91 0.9 0.89 Best Worst Best 930 723 797 7550

0.91 0.9 0.89 Worst Worst Best 99 8909 8 984

0.91 0.9 0.89 Best Best Worst 105 12 9064 819

0.91 0.9 0.89 Worst Best Worst 847 886 768 7499

0.91 0.9 0.89 Best Worst Worst 9068 9 79 844

0.91 0.9 0.89 Worst Worst Worst 8931 85 9 975

Truel SimulationKey:Acc – Player’s accuracy.

Str – Player’s strategy (target best or worst player).

Wins –The number of times (out of 10,000) a player survives.

Strategy in Game Theory

DefinitionA strategy function maps every game

state to an action to take.For a game with finite states, one can

program a response for every game state.

Strategies for simultaneous truel.-Shoot the most accurate opponent.-Shoot the least accurate opponent.

Nash Equilibria

• John Nash (1928- ) developed important game theory concepts.

• Nash equilibrium – an outcome in which no player can do better by changing strategies.

• Every game has at least one Nash equilibrium.

Simulation ResultsAccuracies:

A – 90%

B – 70%

C – 50%

A B

A

A 6.51%

B 3.49%

C 79.98%

A 13.34%

B 1.66%

C 69.97%

C

A 39.23%

B 4.46%

C 15.05%

A 82.50%

B 0.25%

C 1.32%

A B

A

A 3.76%

B 82.12%

C 1.51%

A 17.91%

B 33.91%

C

4.35%

C

A 13.88%

B 53.47%

C 0.31%

A 64.54%

B 3.63%

C 0.04 %

Simulation ResultsNash Equilibria exists when no player can do better by unilaterally changing his or her strategy.

A B

A

A 6.51%

B 3.49%

C 79.98%

A 13.34%

B 1.66%

C 69.97%

C

A 39.23%

B 4.46%

C 15.05%

A 82.50%

B 0.25%

C 1.32%

A B

A

A 3.76%

B 82.12%

C 1.51%

A 17.91%

B 33.91%

C

4.35%

C

A 13.88%

B 53.47%

C 0.31%

A 64.54%

B 3.63%

C 0.04 %

Accuracies:

A – 90%

B – 50%

C – 30%

Simulation Results

A B

A

A 22.46%

B 3.60%

C 63.22%

A 32.82%

B 1.88%

C 50.63%

C

A 56.19%

B 2.65%

C 15.24%

A 83.30%

B 0.21%

C 1.45%

A B

A

A 14.89%

B 66.19%

C 1.69%

A 30.47%

B 36.39%

C 2.53%

C

A 33.10%

B 33.50%

C 0.39%

A 63.47%

B 3.48%

C 0.08%

Simulation Conclusions

These are the only 3 possible equilibria. If both opponents shoot at you, you’ll

want to shoot the better one firstIf your opponents shoot at each other,

you’ll still want to shoot the better one first.

We can eliminate five outcomes based on this logic.

Further Examples: Finite Bullets

• 3 Players: A, B, and C (Original Rules Apply)– None are perfect shots. P(A) = 100% Bullets: 1P(B) = 70% Bullets: 2P(C) = 30% Bullets: 6– No shooting in the air.– Simultaneous

Markov ChainsA has 1 bullet and shoots at B

B has 2 bullets and shoots at A

C has 6 bullets shoots at A

Each player shoots their most accurate opponent

1T (A,B,C) (A,B, —) (A, —,C) (—,B,C) (A, —,—) (—, B, —) (—,—, C) (—,—,—)

(A,B,C) cba qqq 0 cba qqp )1( baa qqq 0 0 )1( cba qqp 0

(A,B, —) 0 baqq 0 0 baqp ba pq 0 ba pp

(A, —,C) 0 0 caqq 0 caqp 0 ca pq ca pp

(—,B,C) 0 0 0 cbqq 0 cbqp cb pq cb pp

(A, —,—) 0 0 0 0 1 0 0 0 (—,B, —) 0 0 0 0 0 1 0 0 (—,—,C) 0 0 0 0 0 0 1 0 (—,—,—) 0 0 0 0 0 0 0 1

Markov ChainsPlayer A has no more bullets

Player B has 1 bullet left

Player C has 5 bullets left

(A,B,C) (A,B, —) (A, —,C) (—,B,C) (A, —,—) (—, B, —) (—,—, C) (—,—,—)

(A,B,C) cbqq 0 0 cbqq1 0 0 0 0

(A,B, —) 0 bq 0 0 0 bp 0 0

(A, —,C) 0 0 cq 0 0 0 cp 0

(—,B,C) 0 0 0 cbqq 0 cbqp cb pq cb pp

(A, —,—) 0 0 0 0 1 0 0 0 (—,B, —) 0 0 0 0 0 1 0 0 (—,—,C) 0 0 0 0 0 0 1 0 (—,—,—) 0 0 0 0 0 0 0 1

2T

Markov ChainsPlayer A has no more bullets

Player B has no more bullets

Player C has 4 bullets left

(A,B,C) (A,B, —) (A, —,C) (—,B,C) (A, —,—) (—, B, —) (—,—, C) (—,—,—)

(A,B,C) cq 0 0 cp 0 0 0 0

(A,B, —) 0 1 0 0 0 0 0 0 (A, —,C) 0 0 cq 0 0 0 cp 0

(—,B,C) 0 0 0 cq 0 0 cp 0

(A, —,—) 0 0 0 0 1 0 0 0 (—,B, —) 0 0 0 0 0 1 0 0 (—,—,C) 0 0 0 0 0 0 1 0 (—,—,—) 0 0 0 0 0 0 0 1

3T

Markov ChainsPlayer A has 1 bullet and 100% accuracy

Player B has 2 bullets and 70% accuracy

Player C has 6 bullets and 30% accuracy

Each player shoots the most accurate opponent.

(A,B,C) (A,B, —) (A, —,C) (—,B,C) (A, —,—) (—, B, —) (—,—, C) (—,—,—) (A,B,C) 0 0 1.21% 0 0 0 98.79% 0

(A,B, —) 0 0 0 0 30% 0 0 70% (A, —,C) 0 0 0 0 70% 0 0 30% (—,B,C) 0 0 0 0.11% 0 61.45% 12.10% 38.79% (A, —,—) 0 0 0 0 1 0 0 0 (—,B, —) 0 0 0 0 0 1 0 0 (—,—,C) 0 0 0 0 0 0 1 0 (—,—,—) 0 0 0 0 0 0 0 1

Markov ChainsPlayer A has 1 bullet and 100% accuracy

Player B has 2 bullets and 70% accuracy

Player C has 6 bullets and 30% accuracy

Each player shoots the opponent with the most bullets.

(A,B,C) (A,B, —) (A, —,C) (—,B,C) (A, —,—) (—, B, —) (—,—, C) (—,—,—) (A,B,C) 0 15% 0 0 50% 35% 0 0

(A,B, —) 0 0 0 0 30% 0 0 70% (A, —,C) 0 0 0 0 50% 0 0 50% (—,B,C) 0 0 0 0.14% 0 40.25% 19.36% 40.25% (A, —,—) 0 0 0 0 1 0 0 0 (—,B, —) 0 0 0 0 0 1 0 0 (—,—,C) 0 0 0 0 0 0 1 0 (—,—,—) 0 0 0 0 0 0 0 1

Absorbing Markov Chains•Absorbing Markov Chains have states that once entered cannot be changed.

Canonical form of a Markov ChainWe can see in the matrix below how the transition matrix is built from four other matrices.

Q is the matrix that represents probability of transition from one transitive state to another

R is the matrix representing the possibility of changing from a transitive state to an intransitive state.

0 is the zero matrix and I is the Identity matrix.Transitive Intransitive

Transitive

Q R

Intransitive

0 I

Absorbing Markov Chains

Canonical form of a Markov ChainThere exists a Fundamental Matrix, N, which gives us the expected number of times the process is in a transient state given an initial transient state.

This is derived by taking the inverse of (I – Q), which is the infinite geometric series in matrix form.

By multiplying this by R, we can find the probability of entering an intransitive state from a transitive state.

This is exactly what we need to fill a payoff matrix.Transitive Intransitive

Transitive

Q R

Intransitive

0 I

Absorbing Markov ChainsLet’s take matrix T1 from earlier, where all players had bullets, and assume they all have infinite bullets.

Then we break it down into our submatrices:

Q R

Absorbing Markov ChainsWe’ll apply the accuracies

A – 80%

B – 50%

C – 30%

5385.1000

01628.100

001111.10

2151.03501.001.0735

)( 1QIN

RQ

NR

Complexity of Markov chains

• We want to find the number of cells which aren’t guaranteed to be zero or one.

• Certain states can’t be reached from others.• Any nuel devolves into an (n-1)-uel• Using counting techniques we find the expression

n

i

i

i

n

2

2!

!

•For n of at least 6, this is approximated by

!39.4)3(! 2 nen

Conclusions for Markov Chains

Markov chains are better for mathematical analysis.

They are not as easily computer-generated as computer simulations.

They have some patterns which may yield more results.

Complexity of Payoff Matrices

• Each Nuel has a payoff matrix which is n dimensions by (n-1)! entries

• This means that a nuel has a payoff matrix with entries.

• Given a set of probabilities, we are not sure if this is NP complete.

nn )!1(

Future Research

• Investigate patterns in R and Q matrices

• Research generating transition matrices

• Find practical applications of theory

Acknowledgements

We would like to give a special thanks toDr. Derado for his guidance during the

project.