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# Decision Maths

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Have you ever wondered how the post office decides what order to deliver mail in? Or how a delivery service can bring everyone on the list a meal without having to backtrack a lot? Have you ever used a flow chart? Then you have already delved into decision maths!

## Applications of decision mathematics

Before looking at the official definition of decision mathematics, let's look at some of the topics included in it.

• Computer algorithm design: Even the basic bubble sort is an example of decision mathematics in computer science.

• Graph theory: You have probably already heard of the classic travelling salesperson problem which involves finding the shortest path between all of the nodes on a graph.

• Linear Programming: Here you will encounter things like the Simplex algorithm, which is used to find a basic solution that satisfies a list of restraints.

• Critical Path Analysis: Anyone who has worked in customer service has encountered critical path analysis, which is used to figure out the quickest way to complete a project given a list of tasks, the relationship between the tasks, and how long each task takes.

• Game Theory: How should you bet on games at a casino to win the most money anyway?

There are plenty more as well!

## Definition of decision maths

So what exactly is decision maths anyway?

Decision maths is a branch of applied probability theory that identifies values and uncertainties related to decisions.

The goal is to make the best decision where each factor in the problem is given a probability of occurring. This can be as simple as deciding not to take an umbrella on your walk because the weather report says there is a low probability of rain today. By not taking an umbrella, you have just made a decision based on probability.

## Types of Decision Mathematics

Decision maths is generally broken down into four types.

1. Making route choices: Package delivery services are especially interested in this type.

2. Influencing outcomes: You do this when you cook a meal and need to decide when to start various dishes based on the goal of having them all finish cooking at the same time.

3. Placing bets: Every professional gambler looks at the odds and probabilities before they place bets. Otherwise, they don't stay professional very long. This isn't just for casino gamblers, stock brokers use this too.

4. Making strategic decisions: Every coach makes strategic decisions, from which players to field to how often to practice.

## Importance and uses of decision mathematics

Decision mathematics is also called operations research. The idea is to use apply analytical methods to help solve problems and make better decisions. This can lead to managers making better decisions on how the business runs so it is more energy efficient.

• Critical Path Analysis: Suppose that a new road is being constructed. People need to be hired, supplies brought in, permits gotten, etc. If you know how long each task should take, and how they are related, it can help you figure out when each task should start, leading to less congestion on alternate paths while the new road is built.

• Bin Packing: Every shipping company deals with bin packing, whether it is container ships, airlines squashing your luggage into a cargo hold, or you moving to a new home. Being able to pack efficiently given the constraints of things like shape, weight, and size, can lead to enormous fuel savings and a cleaner environment.

• Matching Problem: Have you taken a chemistry class? Then you know about covalent bonds. Matching problems are used for things like modelling the bonds in chemistry.

• Route Inspection Problem: Over the river and through the woods, to Grandma's house you go! You have just inspected and chosen a route. If you use GPS apps to help you find directions, then you have worked with route inspection problems.

## Decision mathematics examples

You have already seen several examples of the ways decision maths is used, but let's look at another.

Dynamic programming might sound like a way to have a really exciting event, but it is really a way to optimize things like recursion. Recursive solutions to problems involve repeated calls in a program using the same inputs and are used in things like calculating Fibonacci numbers. Using dynamic programming to decide when and how to save the results of these recursive calls can drastically reduce the time required for a program to run.

This isn't just a programming question though, this is a maths one as well. Have you ever needed to multiply two matrices together? How about three? What if you had to multiply together 150 matrices that were each 3000 by 3000 square? You would certainly want the fastest way to do that! Matrix chain multiplication is another example of the kinds of problems that dynamic programming can help you solve.

## Decision Maths - Key takeaways

• Decision maths is a branch of applied probability theory.
• Applications of decision maths include computer algorithm design, route analysis, bin packing, and critical path analysis.
• Decision maths can be broken down into 4 main types:
• making route choices;
• influencing outcomes;
• making bets; and
• making strategic decisions.

## Frequently Asked Questions about Decision Maths

Dijkstra's algorithm calculates the shortest path between nodes in a graph.

The are graphs where the parts of the tree represent decisions, rules, and outcomes.

Decision maths, also known as operations research, is the study of using analytical models to make better decisions.

Decision maths is used to determine the decidability of a problem. For example, what is the most efficient route to deliver packages, and how to find it quickly.

No. Discrete math applies to a wide variety of topics, including difference equations. Decision math, also known as operations research, is the study of using analytical models to make better decisions.

Decision maths is the study of using analytical models to make better decisions.

## Decision Maths Quiz - Teste dein Wissen

Question

In a two-player, zero-sum game, if player one's pay-off is 4, what is player two's pay-off?

-4.

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Question

In a two-player, zero-sum game, if player two's pay-off is 2, what is player ones's pay-off?

-2.

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Question

What is the value of a two-player, zero-sum game?

The expected value of player one's pay-off, if both players act rationally.

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Question

Given a pay-off matrix for player one, how would you find the corresponding pay-off matrix for player two?

Flip the axis, and multiply all of the numbers in the matrix by -1.

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Question

What is a stable solution of a two-player zero-sum game?

An outcome in which neither player's pay-off would be increased by changing their strategy, whilst the other player's strategy stays the same.

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Question

Do all zero-sum games have a stable solution?

No.

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Question

What do players do if there is no stable solution?

Use a mixed strategy, meaning they play different strategies at random with a certain probability for each choice.

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Question

Can a single game in a rugby tournament be modelled as a two-player, zero-sum game?

Yes.

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Question

Can the Prisoner's Dilemma be modelled as a zero-sum game?

No.

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Question

What is a play-safe strategy?

The strategy with the highest minimum possible pay-off.

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Question

In a two-player, zero-sum game, what is player one's play-safe strategy?

The max(row minimum's), called the maximin.

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Question

In a two-player, zero-sum game, what is player two's play-safe strategy?

The min(row maximums), called the minimax.

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Question

When does a stable solution exist?

When the minimax and maximin values are equal.

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Question

What is the value of a game with a stable solution?

The same as the minimax and maximin values.

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Question

If no stable solution exists, what do we know about the value of the game?

It is bounded between the minimax and maximin values.

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Question

What are algorithms?

An algorithm shows the order in which a process should be followed for an event to occur or for a problem to be solved.

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Question

List the properties of algorithms

• Input
• Output
• Finitness
• Definiteness
• Effectiveness

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Question

How or where can algorithms be applied?

• Algorithms can be used for solving mathematical problems.
• They can be used for solving scientific problems.
• They can be applied in our everyday lives like using a recipe for a meal.
• Computer programmers use algorithms to state out the instructions to follow before writing their codes.

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Question

An algorithm can have more than one input.

TRUE OR FALSE

True

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An algorithm can have more than one output.

TRUE OR FALSE

False

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Question

An algorithm can have zero input.

TRUE OR FALSE

True

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An algorithm should be made to go on and on without stopping.

TRUE OR FALSE

False

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An instruction in an algorithm should have a definite meaning.

TRUE OR FALSE

True

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Question

If an algorithm does not perform the task it is written for, we say the algorithm is...

Ineffective.

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Question

What does it mean if strategy P dominates strategy Q?

Strategy P dominates strategy Q if for every possible strategy the other player would play, playing strategy P results in a higher pay-off than playing strategy Q.

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Question

What can you do with strategy Q if it is dominated by another strategy?

Remove strategy Q from the pay-off matrix.

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Question

If player one has two strategies, A and B, and A dominates B, which of the following statements are true?

All values in row A are greater than or equal to the corresponding values in row B.

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If player two has two strategies, X and Y, and X dominates Y, which of the following statements are true?

The values in the pay-off matrix in column X are all less than or equal to the corresponding values in column Y.

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Question

What is reducing a pay-off matrix?

The action of removing all dominated strategies from the pay-off matrix.

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Question

When can you find a mixed strategy graphically?

When one of the players has only two strategies to use.

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Question

What is a mixed strategy?

A strategy that uses different strategies at random, with a set probability for each strategy.

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Question

At what point is the optimal allocation for the probability p, of choosing strategy A?

When the minimum possible pay-off is maximised.

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Question

Which players perspective must you formulate the Linear Programming problem from?

Player two.

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Question

What is $$x_i$$ defined as?

$x_i = \frac{p_i}{V'}$ where $$p_i$$ is the probability that player one will choose strategy $$i$$, and $$V'$$ is the value of the altered pay-off matrix.

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Question

What is $$y_i$$ defined as?

$y_i = \frac{q_i}{V'}$ where $$q_i$$ is the probability that player two will choose strategy $$i$$, and $$V'$$ is the value of the altered pay-off matrix.

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Question

How can you tell that a tableau does not need to be pivoted anymore?

The top row is all non-negative.

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If no stable solution exists, what sort of strategy do the players use?

A mixed strategy.

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Question

Which of the following statements is true for the following pay-off matrix?

Strategy X dominates strategy Y.

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Question

If each player has 2 strategies and you have found the optimal mixed strategy for player one, how can you find the optimal mixed strategy for player two?

Rewrite the pay-off matrix from player two's perspective and solve in the usual way.

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Question

If player one has 2 strategies and player two has 3 strategies, and you have found player one's optimal mixed strategy, how can you find the optimal mixed strategy for player two?

Create simultaneous equations using the probabilities that player one chooses each strategy, and solve for p, q and (1-p-q).

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Question

How can you solve a game where both players have more than 2 strategies (assuming the pay-off matrix is already fully reduced)?

Using Linear Programming.

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Question

What is a linear programming problem?

A linear programming problem deals with optimising (maximising or minimising) a function.

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What are the three constituents of a linear programming problem?

Objective function, decision variable and constraints.

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Question

Is happiness a valid quantity for a linear programming problem?

No. Only quantifiable quantities elements can be included in a linear programming problem.

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Question

Can the expression $$x^2$$ be an objective function in a linear programming problem?

No. Objective function and constraints in a linear programming problem must be linear equations or inequalities.

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Question

A firm manufactures two types of products, A and B and sells them at a profit of $2 on type A and$3 on type B. What would be the objective of a linear programming problem which has to maximise the profit?

Maximise $$2x_1+3x_2$$

Here, $$x_1$$ denotes the units of product A and $$x_2$$ the units of product B.

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Question

A firm manufactures two types of products, A and B and sells them at a profit of $2 on type A and$3 on type B. What are the decision variables of a linear programming problem which has to maximise the profit?

The firm has to decide how many units of products A and B are to be manufactured to maximise its profit. So, they are the decision variables.

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Question

A person wants to decide the constituents of a diet which will fulfill his daily requirements of proteins, fat, and carbohydrates at the minimum cost. The choice is to be made from four different types of foods. What are the decision variables for this optimisation problem?

The units of food of type 1, 2, 3, and 4 to be consumed to maintain the desired diet are the decision variables.

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Question

What is the function to be optimised in a linear programming problem called?

Objective function

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Question

David wants to optimise his travel route from his house to work. There is only one roadway connecting his house and workplace. Can this be formulated as a linear programming problem?

No. There must be alternative courses of the line of action to choose from.

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