Hunger Games Combat Model Vany Nguyen, Margaret Tian, Wanlin Xie Introduction Our simulation will be based off the book series The Hunger Games. In the first book, the main character, Katniss, battles the tributes of other districts but also forms alliances. We are going to simulate the battles with 6 tributes, 3 people in each alliance. The alliances do not have equal power, since each individual has a different skill level. The first alliance is Katniss, Peeta, and Rue. Their individual skill levels are 11, 8, and 7, respectively. The second alliance is Cato, Clove, and Marvel. Their skill levels respectively are 10, 10, and 9. To make this model easier, we added up the individual score and took it out of 36, the highest score possible for an alliance of 3. Katniss’s team has a score of 26 out of 36 and Cato’s has a score of 29 out of 36. Assumptions We made many assumptions to make this problem solvable. When one person dies on each team, only one third of the team's combined skill level is lost regardless of who dies. So, we treat each person the same as his or her teammates so that who specifically was killed doesn't alter the equations. We also assume that the other tributes do not fight the two alliances or affect them in any way. In addition, we assume a time interval of one day as each alliance will essentially only fight once per day. From just looking at their skill levels, we can see that Cato's team will have a higher probability of winning. S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 2nd Scenario This time, we have taken to account that Katniss' team is weakened due to hunger and lack of shelter while Cato's team has sufficient supplies. Let's see how this affects their probability of winning. To model this, we lower the skill level of Katniss' team from 26 to 20. 3rd Scenario After one person on each team has died (Marvel and Rue), and Peeta is severely injured, Cato's team attacks. Due to their love for each other, injured Peeta, now unable to fight, is considered a liability to Katniss, thus lowering her skill level. Let's see how this works out. 1st Scenario In this first scenario, we are just pitting Team Cato against Team Katniss, and seeing the probability of winning for each team. Right now, we are sticking with our original assumptions where each team member is counted the same on each team. Later, when we complicate the question more, we will take into account their individual skill levels. Looking at the graph of the probability of all stages, we can see that the games will end after approximately 4 days; the probabilities of the intermediate stages decrease to basically 0 after 4 days. Blue: Cato (0.548)

Red: Katniss (0.452) Conclusion Through these three scenarios, we can see that Katniss's team is not likely to win, especially when the team is weakened or injured. Although Katniss had a skill level of 11, higher than any individual, she cannot easily bring the team to victory. As predicted, Cato's team has a higher probability of winning in each scenario, increasing every time Katniss's team weakens. However, our results are merely numerical statistics while The Hunger Games is a story, full of camaraderie, emotional drama, and young love, all factors that are impossible for us to model. Even though the odds are not in her favor, Katniss still manages to emerge triumphant in the Games. We used a stochastic model to most accurately represent this scenario. For each stage depicted in the matrix, we wrote an equation showing the flow of its probability Our Results: Blue: Cato (0.659)

Red: Katniss (0.341) Blue: Cato (0.80)

Red: Katniss (0.20) For this scenario, we wrote new equations to model the different conditions and again used Euler's method to solve for the probabilites. We used Euler's method to solve our equations We can see that just by changing Katniss' team's

skill level by a small amount, the result probabilities change drastically. Cato's team's chance of winning as compared to Katniss' rises from 5:4 to almost 2:1. Clearly, food and shelter are large factors in the Hunger Games. Skill Levels 11 8 7 10 9 10 26 0 1 Number of People on Katniss' team 29 2 3 1 2 3 0 Number of People on Cato's team Katniss Victory

Cato Victory As we can see, our predictions were correct. Cato's team has a higher chance of winning. Results: The odds are still in Cato's team's favor, and this time, they have an even higher chance of winning. In this situation, Cato's team has a significantly higher chance of winning, in fact, 4 times higher. With Peeta serving as a handicap, it's very difficult for Katniss to win. Luckily, in the movie, Peeta is healed before Cato's team attacks, or else the Hunger Games may had a different outcome. Totals: BUT WAIT What if Katniss won in our model? 4th Scenario So, in the beginning, Cato's alliance acquired a large amount of supplies. Their strategy was to divide up the supplies so that the opposing team would not destroy them all at once. For every pile of supplies that Katniss finds, Cato's team loses one skill level and Katniss gains one skill level. How many would Katniss have to find to have a 60% chance of winning? So, after plugging in values into our mathcad system, we can see that Katniss needs to find 5 supply piles in order to have at least a 60% chance of winning. As we can see, many different aspects of the Hunger Games can be manipulated to get certain results. The probability of a team winning depends on the skill level which can be affected by a multitude of factors including injury, supplies, alliances, natural disasters, and of course emotions. THANKS FOR LISTENING!

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# Hunger Games Combat Model

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