Prologue

1
Prologue

• http//www.youtube.com/watch?vXP9cfQx2OZY

2
Truels and N-uels

• An Analysis

3
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.

4
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.

5
An Example and Previous Research
Lets consider one situation in Truels and Nuels
- each player is a perfect shot - each shoots
in a sequence.
A
B
C
6
An Example and Previous Research
Taking Turns From 1 Players Perspective 1st
Shooter A A shoots B C can then shoot at A.
A
B
C
7
An Example and Previous Research
However, if A shoots into the air, Bs response
should be to eliminate his only threat, C.
A
B
C
8
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.

9
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 ?

A
?
B
C
10
More Interesting Example

• 3 Players A, B, and C (Original Rules Apply)
• None are perfect shots.
• P(A) 90
• P(B) 70
• P(C) 50
• No shooting in the air.
• Simultaneous
• Results can range from none survive to all
  survive.
• A, B, C, A,B, B,C, A,C,
  A,B,C, Empty Set
• More on this later.

11
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

12
Truel Simulation
Key Acc Players accuracy. Str Players
strategy (target best or worst player). Wins The
number of times (out of 10,000) a player survives.
Acc1 Acc2 Acc3 Str1 Str2 Str3 Wins1 Wins2 Wins3 No
ne 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 Be
st 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 Wor
st 8931 85 9 975
13
Strategy in Game Theory

• Definition
• A 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.

14
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.

15
Simulation Results
Accuracies A 90 B 70 C 50
Survival Probabilities in a Simultaneous Truel
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
C
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
B shoots
A shoots
B
C shoots
16
Simulation Results
Nash Equilibria exists when no player can do
better by unilaterally changing his or her
strategy.
Survival Probabilities in a Simultaneous Truel
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
C
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
B shoots
A shoots
B
C shoots
17
Simulation Results
Accuracies A 90 B 50 C 30
Survival Probabilities in a Simultaneous Truel
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
C
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
B shoots
A shoots
B
C shoots
18
Simulation Conclusions

• These are the only 3 possible equilibria.
• If both opponents shoot at you, youll want to
  shoot the better one first
• If your opponents shoot at each other, youll
  still want to shoot the better one first.
• We can eliminate five outcomes based on this
  logic.

19
Further Examples Finite Bullets

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

20
Markov Chains
A 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
21
Markov Chains
Player A has no more bullets Player B has 1
bullet left Player C has 5 bullets left
22
Markov Chains
Player A has no more bullets Player B has no more
bullets Player C has 4 bullets left
23
Markov Chains
Player 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.
24
Markov Chains
Player 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
25
Absorbing Markov Chains

• Absorbing Markov Chains have states that once
  entered cannot be changed.
• Canonical form of a Markov Chain
• We 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
26
Absorbing Markov Chains
Canonical form of a Markov Chain There 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
27
Absorbing Markov Chains
Lets 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
28
Absorbing Markov Chains
Well apply the accuracies A 80 B 50 C 30
29
Complexity of Markov chains

• We want to find the number of cells which arent
  guaranteed to be zero or one.
• Certain states cant be reached from others.
• Any nuel devolves into an (n-1)-uel
• Using counting techniques we find the expression

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

30
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.

31
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.

32
Future Research

• Investigate patterns in R and Q matrices
• Research generating transition matrices
• Find practical applications of theory

33
Acknowledgements

• We would like to give a special thanks to
• Dr. Derado for his guidance during the project.