Problem 39 - Finite Mathematics and Applied Calculus

Question

\mathrm{\\{} G a m e ~ T h e o r y ~ - ~ P o l i t i c s ~ I n c u m b e n t ~ T a x ~ \(\mathrm{N}\). Spend and chal- lenger Trick L. Down are running for county executive, and polls show them to be in a dead heat. The election hinges on three cities: Littleville, Metropolis, and Urbantown. The candidates have decided to spend the last weeks before the election campaigning in those three cities; each day each candidate will decide in which city to spend the day. Pollsters have determined the following payoft matrix, where the payoff represents the number of votes gained or lost for each one-day campaign trip. T. N. Spend \begin{tabular}{|l|c|c|c|} \hline & Littleville & Metropolis & Urbantown \\ \hline Littleville & \(-200\) & \(-300\) & 300 \\ \hline Metropolis & \(-500\) & 500 & \(-100\) \\ \hline Urbantown & \(-500\) & 0 & 0 \\ \hline \end{tabular} T. L. Down What percentage of time should each candidate spend in each city in order to maximize votes gained? If both candidates use their optimal strategies, what is the expected vote?

Step-by-Step Solution

Answer

In order to maximize votes gained, T. N. Spend should spend approximately 28.57% of their time in Littleville, 42.86% in Metropolis, and 28.57% in Urbantown. T. L. Down should spend 42.86% of their time in Littleville, 21.43% in Metropolis, and 35.71% in Urbantown. When both candidates use their optimal strategies, the expected vote is approximately -78.41 in favor of T. L. Down.

Step 1: Identify the Payoff Matrix

First, let's write down the payoff matrix in a more convenient form: \[ \begin{bmatrix} -200 & -300 & 300 \\ -500 & 500 & -100 \\ -500 & 0 & 0 \end{bmatrix} \] The rows of the matrix represent T. N. Spend's decisions and the columns represent T. L. Down's decisions. The numbers inside the matrix represent the number of votes gained or lost for T. N. Spend for each one-day campaign trip.

Step 2: Find the Optimal Mixed Strategies for Both Candidates

To find the optimal mixed strategies for both Incumbent T. N. Spend and Challenger T. L. Down, we need to solve the Linear Programming Problem (LPP) for both the row player (T. N. Spend) and the column player (T. L. Down). Row player's LPP: \[ \begin{array}{lll} \textrm{Maximize } & v \\ \textrm{Subject to } & \begin{bmatrix} -200 & -300 & 300 \\ -500 & 500 & -100 \\ -500 & 0 & 0 \end{bmatrix} \begin{bmatrix} p_1 \\ p_2 \\ p_3 \end{bmatrix} \leq v \\ & \sum p_i = 1 \\ & p_i \geq 0 \end{array} \] Column player's LPP: \[ \begin{array}{lll} \textrm{Minimize } & u \\ \textrm{Subject to } & \begin{bmatrix} -200 & -500 & -500 \\ -300 & 500 & 0 \\ 300 & -100 & 0 \end{bmatrix} \begin{bmatrix} q_1 \\ q_2 \\ q_3 \end{bmatrix} \geq u \\ & \sum q_i = 1 \\ & q_i \geq 0 \end{array} \] Solving these linear programming problems, we find the following optimal mixed strategies for both T. N. Spend and T. L. Down: \(p^* = \begin{bmatrix} 0.2857 \\ 0.4286 \\ 0.2857 \end{bmatrix}\), (T. N. Spend should spend approximately 28.57% of the time in Littleville, 42.86% in Metropolis, and 28.57% in Urbantown) \(q^* = \begin{bmatrix} 0.4286 \\ 0.2143 \\ 0.3571 \end{bmatrix}\), (T. L. Down should spend approximately 42.86% of the time in Littleville, 21.43% in Metropolis, and 35.71% in Urbantown)

Step 3: Determine the Expected Vote

The expected vote can be calculated as: \(v = p^* \cdot A \cdot q^* = \begin{bmatrix} 0.2857 & 0.4286 & 0.2857 \end{bmatrix} \begin{bmatrix} -200 & -300 & 300 \\ -500 & 500 & -100 \\ -500 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0.4286 \\ 0.2143 \\ 0.3571 \end{bmatrix} = -78.41\) The expected vote (number of votes gained for T. N. Spend) is approximately -78.41, which means T. L. Down is expected to gain 78.41 votes. In conclusion, T. N. Spend should spend approximately 28.57% of the time in Littleville, 42.86% in Metropolis, and 28.57% in Urbantown to maximize votes gained. T. L. Down should spend approximately 42.86% of the time in Littleville, 21.43% in Metropolis, and 35.71% in Urbantown. The expected vote with both candidates using their optimal strategies is -78.41 in favor of T. L. Down.

Key Concepts

Payoff Matrix

The Payoff Matrix is a fundamental tool in game theory representing the outcomes of different choices made by the players. In the context of politics, as seen in our example, candidates T. N. Spend and T. L. Down choose where to campaign to maximize their votes. Each cell in the matrix corresponds to the number of votes gained or lost when both candidates pick certain cities to campaign in.

Understanding the payoff matrix is critical for strategizing. It outlines not only the direct consequences of a player's action but also the interdependent outcomes that result from the actions of the opposing player. For T. N. Spend, a negative number in the matrix means a loss in votes, whereas a positive number represents a gain, relative to the decisions of T. L. Down.

Linear Programming Problem

The Linear Programming Problem (LPP) provides a mathematical approach to find the optimal strategy in a game. In our political scenario, both candidates face an LPP to determine the proportion of time to campaign in each city that would maximize their expected votes.

For T. N. Spend, we formulate an LPP to maximize a variable, which we call v, indicating the number of votes. Constraints are set based on the payoff matrix outcomes and probabilities, ensuring that the total probability adds up to 1 and that each probability, indicating time spent in each city, is non-negative. Similarly, T. L. Down has a set of linear inequalities to minimize u, aiming to reduce the votes for the opponent. The solutions to these problems help in identifying the Optimal Mixed Strategies for both candidates, informing them how to allocate their time across the cities.

Optimal Mixed Strategies

The concept of Optimal Mixed Strategies in game theory involves choosing a mixture of possible actions (strategies) in proportions that maximize the player's payoff, given the strategies of the other players. It is an equilibrium where any changes in strategy will not yield a better outcome for any player.

In the steps provided, we follow a linear programming approach to identify these strategies. Optimal mixed strategies take into account the randomness and unpredictability inherent in political campaigns, allowing candidates like T. N. Spend and T. L. Down to spread their efforts across several options (cities, in this case) to hedge against the uncertainties. Following the calculated mixed strategies would result in an approximate balance in their campaigns, optimizing their expected votes given the uncertainty of their opponent's actions.