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Prologue http://www.youtube.com/watch?v=XP9cfQ x2OZYhttp://www.youtube.com/watch?v=XP9cfQ x2OZY

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Truels and N-uels An Analysis

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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.

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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.

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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.

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An Example and Previous Research Taking Turns From 1 Player’s Perspective 1 st Shooter: A A shoots B C can then shoot at A.

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An Example and Previous Research However, if A shoots into the air, B’s response should be to eliminate his only threat, C.

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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.

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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 ?

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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.

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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

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Acc1Acc2Acc3Str1Str2Str3Wins1Wins2Wins3None 0.910.90.89BestBestBest9819146764 0.910.90.89WorstBestBest9910288801 0.910.90.89BestWorstBest9307237977550 0.910.90.89WorstWorstBest9989098984 0.910.90.89BestBestWorst105129064819 0.910.90.89WorstBestWorst8478867687499 0.910.90.89BestWorstWorst9068979844 0.910.90.89WorstWorstWorst8931859975 Truel Simulation Key: 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.

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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.

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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.

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Simulation Results Accuracies: A – 90% B – 70% C – 50% AB 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% AB 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 %

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Simulation Results Nash Equilibria exists when no player can do better by unilaterally changing his or her strategy. AB 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% AB 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 %

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Accuracies: A – 90% B – 50% C – 30% Simulation Results AB 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% AB 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%

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Simulation Conclusions These are the only 3 possible equilibria. If both opponents shoot at you, you’ll want to shoot the better one first If 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.

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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

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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

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Markov Chains Player A has no more bullets Player B has 1 bullet left Player C has 5 bullets left

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Markov Chains Player A has no more bullets Player B has no more bullets Player C has 4 bullets left

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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.

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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.

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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. TransitiveIntransitive TransitiveQR Intransitive0I

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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. TransitiveIntransitive TransitiveQR Intransitive0I

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Absorbing Markov Chains Let’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:

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Absorbing Markov Chains We’ll apply the accuracies A – 80% B – 50% C – 30%

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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 For n of at least 6, this is approximated by

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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.

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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.

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

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

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