# application of linear algebra in games

Below is an example of a payoff matrix for a game of Rock, Paper, Scissors between Player A and Player B: In this matrix, player A is playing the rows, while player B is playing the columns. Doing linear regression is trying to solve Ax = b. In our game, we have 25 lights to keep The math looks like this: In this case, λ, our eigenvalues, are equal to 0.6 and 1. Suppose that this process continues for multiple years. to any kind of region or area. What would the results be? Then create your own network with flows and find the steady state of the system. Do not worry however, because will be using a modified version of the "rref()" Suppose you are the offensive coordinator of a football team, and you run 5 main plays. A python version of the original game can be found below. For the first four projects at least two team members must have experience with programming. rooms that correspond to 1 in the vector we need to perform row reduction on the augmented matrix [R In order to simulate an actual epidemic, I wrote a short Python program using the additional modules NumPy (to do matrix math) and matplotlib (to plot the data) to iterate through the linear function for a range of 200 days. This is done for every eigenface used in the comparison. of linear algebra in the most unexpected of situations. Resources This allows us to plug That would certainly make the princess happy! switch affects that state. Our starting population can be represented by a vector (similar to the example from before) where everyone is disease free: So what happens on day 1 of the zombie outbreak? The rest of the susceptible population remains so. If a system has eigenvalues that are all negative, the equilibrium point would be a sink, or all solutions would head towards that point, as water heads down a drain. So in this scenario, in a normally functioning economy, the ratio of employed to unemployed workers should be 3 to 2. sfinal] and read off the What we get on the right hand side is the First, we want to find the eigenvalues, and then a vector that satisfies Av=v. V = voltage, I = Current, and R = Resistance. Also, An is equal to PDnP-1. The more variance a face has, then less eigenfaces are needed. corresponding to this choice of free variables, is just the last column The factor on the left is the number of times we From this equation, we can determine the eigenvectors (Eλ) for the eigenvalues by solving for the free variables. This is done by first checking the rows. To find the eigenvalues of this system, we find : So our eigenvalues for this system are , , and . A Free and Open-Source Linear Algebra Text Mathematics is for everyone – whether as a gateway to other ﬁelds or as background for higher level mathematics. with a current of I1 through it. To see a brief elementary introduction to many of the applications listed below, a good resource is: H. Anton and C. Rorres, “Elementary Linear Algebra Applications”. We see Column 2 dominates Column 1, and so we eliminate Column 1 and are left with: Now, we can again reduce by dominance with rows as Row 2 dominates Row 3 and we are left with: We are left with just one row, row 2, or Play 2. If we multiply the population vector by the transformation matrix, we can find the changes in the populations as time progresses. Are there any more free variables and so the matrix of coefficients cannot be However, as n approaches large numbers, the second term is still multiplied by a factor of one. We call this image, in its entirety, the Mandelbrot set. There are many different types of game theory problems out there, but we are going to look at a simple one that is the foundation of modern game theory. It will say something like "Save + List them. solution. to the following choices of the pair (ε24, we will have 25 possible operations, one corresponding to each room. You continue this process through each loop in If you are working from home, save the file to whatever you are using A poor 21% will recover but. Given a matrix A, the eigenvalues can be found by solving for lambda in the equation det(A-λI)=0. They minimize the distance e between the model and the observed data in an elegant way that uses no calculus or explicit algebraic sums. We can reduce by row dominance, so we then turn to the columns. zero vector, we can rewrite our equation as: ε1R1 0 = 0 (This means "not Using the function from above: The disease is official spreading! Therefore, E.6 is equal to the eigenvector , which is equivalent to . Programming is used to implement those algorithms, the algorithms are not used for general programming. That is, starting from the initial state, we flip the In order to find which points are in the set, we put the real and complex numbers into a matrix and simplify, which allows us to find the behavior of our equations. Applications of Linear Algebra. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T. Now we need to introduce notation corresponding We use the equation. To put this in Matrix form, we will create a vector containing and and call it , and we will create the transition matrix with the employment probabilities. A covariance matrix determines how much a set of data varies from the mean. concern the operations on the lights. This means that a positive number indicates a win of that amount by Player A (and loss for Player B), and a negative number indicates a loss for Player A (and win for Player B). (If there When x = 3, b = 2 again, so we already know the three points don’t sit on a line and our model will be an approximation at best. save the file to MATLAB's working directory. Here is how it works. As you can see with our application, basic linear algebra concepts can be useful in analyzing systems in upper-level math classes. Since sinit is just the The first entry is Philadelphia’s population and the second is Pittsburgh’s: To show the population shifting between cities, we create a transformation matrix using the percentage shifts: In the first column, we see that 70% of Philadelphia’s population stays where they are, while 30% moves to Pittsburgh. The idea is to store the state of 25 lights in a This key is unique to this door and can only open this door. get us from sinit to sfinal. download the file rrefmod2.m. Of the zombies, 7% will die. R25. sfinal] using the arithmetic operations Looking at the matrix, we see that row 2 dominates rows 1, 4, and 5, so they are all eliminated from the matrix and we are left with: This means that for the offensive coach, we have eliminated Plays 1, 4, and 5 and now have to decide between Play 2 or Play 3. For the purpose of our example, everyone else stays where they are. This is a result of having a y-value of 1. It takes This is determined by the uniqueness of each person’s face. in a matrix consisting of zeros and ones and it performs the function is a free variable, remember that it can be only 0 or 1, so for every Through using eigenvalues and determinants, we can analyze how systems behave over time. gives us the equations I1+I2+…+In = 0 and V0 A single shape, such as a triangle, will be made up of infinitely many other triangles. Beware that this is not a standard MATLAB command, but it was created Labor market statistics are very important for determining the well being of a country. against I1’s loop. Mountain ranges, ocean waves, snow flakes, lightning bolts, crystals, and even pineapples are composed of fractals! Once all the vectors are in place, the mean of all of the vectors is taken. This means we won't touch the light switches in rooms R24 and But things don’t happen the same way when we use the third point. command to row reduce our matrix R. This command is In the previous section we were able to create a Resources "The math behind these games", source as .zip; Sweep the board (Python version by Qirong Li) We see that solving the puzzle reduces to solving column vector. We can do this using : The above equation represents the general characteristic polynomial of matrix . 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0). ε24 = ε25 = 1. Objects such as our phones, laptops, and home computers now have the capability to unlock themselves just by taking a look at the face of the beholder of the device. flipping the switch zero times is the same as not flipping the switch"). the circuit. Especially concerned by this trend is the subject math. Each move is simply the addition of one of the vectors Ri I then plotted the results which are shown below: Breaking down the graph, our model predicts that approximately 60% of the population will, around 75 to 100 days or so, have become immune to the disease and about 40% of the population will have died.

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