Strategy and tactics

SUMMARY: Mathematical concepts and processes can be used to analyze optimal strategies in a variety of situations.

In a competitive situation, such as businesses selling similar products, armies engaged in battle, opponents playing games, oil companies deciding where to drill, and employees bargaining for better salaries, successful outcomes depend on choosing the best plan of action from among a set of strategies to achieve a specific outcome. In many cases, mathematics can be used to analyze the situation and help to choose the best strategy. Mathematical techniques have been—and will continue to be—developed to address a wide range of problems in areas such as military logistics, intelligence, and counterintelligence.

The first step in the process is to determine the objective. That goal may be to maximize profit, beat the opposing army, or win the game. Next, the possible strategies to choose from and the limitations or constraints that may affect the choice of strategy need to be identified.

In competitive situations, the opponent’s choice of strategy must be taken into consideration as well. While there are many examples of systematically analyzing and selecting the “best” strategies throughout history, the twentieth century—especially the World War II era—saw the emergence of operations research as the discipline that explores and develops systematic techniques for making decisions that are the “best” in some sense, usually maximizing profits/benefits or minimizing costs/liabilities.

Decision making can be approached mathematically in a number of ways depending upon the situation involved and the information available.

Linear Programming: Choosing the Best Option When Resources are Limited

Many decision problems arose out of troop supply needs during World War II. With a war on several fronts, deciding how to ship the limited troops and supplies to maximize their effectiveness was daunting. Many of the situations had the following characteristics:

• There were resources needed in specific combinations by a number of end users, and the amount of each resource was limited.
• The resources were used proportionally for each combination (in other words, to assemble whole units from raw materials, the number of raw materials needed was the same for each unit produced).
• The goal was to maximize the benefit or minimize the cost, and the cost or benefit was proportionally related to the number of units produced (in other words, the more produced, the higher the benefit or cost).

These characteristics yield a mathematical structure that is linear. Each resource corresponds to an equation or inequality that is a linear combination of unknown quantities representing the units to be combined or produced The objective function is also a linear combination of the number of units. See Example 1 for a very simple, classic example that involves deciding how to prepare a “balanced” meal.

Example 1. A linear programming problem

A dietician wishes to prepare a salad meal that has a minimum amount of calories but still satisfies nutritional requirements. In particular, it must have at least 30 grams of protein and at most 9 grams of fat. The foods available are an ounce of lettuce with 4 calories, no fat, and 1 gram of protein; and slices of roast beef with 90 calories, 3 grams of fat, and 16 grams of protein. What amounts of lettuce and beef should the dietician serve with a diet salad dressing? Minimize calories =4L+90B, where 1L+16B≥30 and 3B≤9.

These problems are easy to solve when they are small, like the problem in Example 1. The problems that arise in practice—such as those under consideration during World War II—are usually much larger and can involve hundreds of unknowns. During World War II, British and US mathematicians looked for an approach that could make use of computers, which were being developed at that time and offered the possibility of performing many simple calculations quickly. In 1947, too late for the war effort, US mathematician George Danzig (1914–2005) developed the simplex algorithm for solving linear programming problems. The simplex algorithm is an efficient recipe for solving linear programming problems of any size and is very easy to program on a computer. In the decades since the development of the simplex algorithm, many industries have used this procedure to solve problems in fields as diverse as banking, natural resources, manufacturing, and farming.

Linear programming problems are usually used to model static situations in that the final solution is essentially the result of one decision made under a clear set of assumptions. Many decision problems are more complicated, with a number of intermediate decisions to be made. These more dynamic problems often involve a probabilistic component as well, with uncertainty playing a complicating role in each decision.

Game Theory

Often, people are faced with a decision in which the resulting payoff will depend on external forces that are hard to predict (like natural forces). One option may always be best, but it is more likely that the best choice will simply “depend” on other factors. For example, when deciding which crop to plant, a farmer can list seed costs and profits based upon yield, but the yield will depend on the weather. A table can be made for each crop choice based upon several different weather scenarios, with past experience used to assign a probability to each possible weather scenario. Example 2 provides a standard format, usually called the “payoff matrix.”

Example 2. A payoff matrix

List the possible states of the external forcesList the possible actions to choose from in making the decisionList the gain (profit, benefit, etc.) for each combination of actions and states.

Many decisions can be similarly structured, including determining what stocks to buy, what products to market, and what wars to wage. Different people will make different decisions depending upon their comfort level with risk.

Strategies for systematic decision making can be placed in four categories:

1. Optimist strategy: “MaxiMax” (Maximize the maximum gain). Find the best gain for each possible action and choose the largest of these maximums. Of course, that action may have the most risk associated with it, since the maximum gain may also coincide with the least likely state for the external force. In this case, the farmer may plant something that would have huge profits but only in the most unlikely weather conditions.
2. Pessimist strategy: “MaxiMin” (Maximize the minimum gain). Find the smallest gain for each action, and choose the largest of these minimums. This is a safe choice because it yields the minimum guaranteed gain regardless of external forces. In this case, the farmer may choose a “safe” crop to plant. If weather is really good, another crop would have been a better choice.
3. Balanced strategy: “MiniMax Regret.” Calculate the “regret” for each possible action by determining the cost of choosing that action compared to benefits of the best state of the external forces. Find the worst (largest) regret for each action and pick the action with the smallest worst-case regret.
4. Averaging strategy: “Expected Value.” Use the probabilities governing the external forces to determine the expected gain for each action and choose the highest one. Expected gain or payoff is calculated as a weighted average of the gain for each state of the external force where the weight for each state is the probability of that state occurring. This strategy can be thought of as determining the action that, when chosen repeatedly, provides the best average benefit over the long term. For the farmer, this may not seem reasonable, since the decision under consideration is what to plant in a single, given year.

When the external force is an opponent with choices to make rather than a natural phenomenon with a random component, these decision situations can be examined as mathematical games. Two-person games can be represented with a payoff matrix as in Figure 2. The “row” player lists strategies on the left and the “column” player lists strategies across the top. The entries of the matrix are pairs of numbers, the row player’s payoff, and the column player’s payoff, respectively. In situations where the row player’s winnings are equal to the column player’s losses, and vice versa, the payoff matrix entries can be completely defined with one number, conventionally the row player’s payoff. These games are called “zero-sum games” because for a particular pair of strategies, the row player’s payoff and the column player’s payoff, being negatives of each other, sum to zero.

Example 3. The prisoner’s dilemma

Two suspects are arrested by the police. They are each offered the same deal: Confess and receive a reduced sentence. If one confesses and the other does not, the confesser goes free and the other gets a 10-year sentence. If both confess, each gets a five-year sentence. If neither confesses, both get a one-year sentence on reduced charges. Neither prisoner knows what the other will say. What should they do?

ConfessRefuseConfess(-5,-5)(0,-10)Refuse(-10,0)(-1,-1)

While mathematicians have been studying decision making and games of strategy systematically for several centuries, game theory emerged as a recognized mathematical approach to analyzing these decision processes in the 1930s and 1940s through research published by John von Neumann (1903–1957). The “prisoner’s dilemma” (Example 3) was investigated in the 1950s and led to additional interest in the field.

The prisoner’s dilemma captures many interesting features of competitive situations. Analysis shows that the intelligent prisoner should always confess, since the “best” outcome will occur no matter what the other prisoner decides to do: -5 is better than -10 if the other prisoner confesses; 0 is better than -1 if the other prisoner refuses to talk. However, this individual “best choice” results in each prisoner confessing and getting a five-year sentence, whereas if neither confesses, they only get one-year sentences. This feature of competitive behavior and strategies can be thought of as the friction between basing strategic decisions on individual goals or on the common good.

With appropriate choices for the values in the table, these games could model a number of competitive situations, such as two companies trying to determine what price to set for competing products or two armies determining how to wage war.

Decision Trees

In situations where the ultimate decision depends on an intermediate choice, a decision tree can help to organize the information and facilitate a systematic analysis. A company may be ready to bring a product to market and needs to decide whether or not to invest funds up front in a test market exercise. The test market may bring in better information about how to market the product on a larger scale, thus increasing profit, but the cost of the test market exercise would also take away from the profit. An oil drilling company could choose to invest funding in test wells before determining the final drilling location. A university may be trying to hire a senior administrator and could choose to invest funds in a head-hunter search firm.

In all of these situations, the outcomes can be organized into a tree diagram like the one in Figure 1. Each “decision fork” is represented by a square, and each event fork—governed by external, possibly random forces—is represented by a circle. The branches leading from the event forks have probabilities assigned based upon the likelihood that an outcome will occur. Typically, acquiring additional information will result in an increased probability of success (or failure), and so the probabilities of success and failure will be different for different event forks.

Each terminal branch represents a final outcome. If current assets, the cost of the information acquisition, and the gains or losses under success and failure are known, then each terminal branch can be labeled with the net gain (or loss) for that option. Once those values are determined, the tree can be “folded back” through calculating the expected outcomes from the probabilities to determine which decisions to make to maximize the gain.

The decision points and events may include more than two options or outcomes, and there may be more than two decisions to be made before the final outcome, so the tree may have more forks and branches than the one in Figure 1 but the analysis process is the same.

From these trees, the value of the additional information acquired can be calculated. This calculation can assist companies in determining how much they should be willing to pay for that information. Also, the amount of risk a company is willing to assume can be incorporated into the process, allowing companies that are willing to shoulder a larger risk for the (slimmer) chance of a larger gain to include that information into the analysis.

Bibliography

Brams, Steven, et al. "Game Theory." Britannica, 24 Oct. 2024, www.britannica.com/science/game-theory. Accessed 20 Dec. 2024.

Mesterton-Gibbons, M. An Introduction to Game-Theoretic Modeling. Redwood City, CA: Addison-Wesley, 1992.

Raifa, H. Decision Analysis: Introductory Lectures on Choices Under Uncertainty. Reading MA: Addison-Wesley 1968.

Winston, W. L. Operations Research: Applications and Algorithms. 4th ed. Belmont, CA: Brooks Cole-Thompson Learning, 2004.