Question

Suppose you are playing a game in which you and one other person each picks a number between 1 and \(100,\) with the person closest to some randomly selected number between 1 and 100 winning the jackpot. (Ask your instructor to fund the jackpot.) Your opponent picks first. What number do you expect her to choose? Why? What number would you then pick? Why are the two numbers so close? How might this example relate to why Home Depot and Lowes, Walgreens and Rite-Aid, McDonald's and Burger King, and other major pairs of rivals locate so close to each other in many well-defined geographical markets that are large enough for both firms to be profitable?

Short answer

Your opponent is expected to choose 50. You should then choose 49 or 51 for optimal chances of winning. Businesses cluster similarly to target a maximized customer base.

Step-by-step solution

Identify the Game Situation

This is a strategic game where two players pick numbers between 1 and 100. The winner is the one whose number is closest to a random number also between 1 and 100. We need to predict the opponent's thinking and plan accordingly.

Predict the Opponent's Strategy

The opponent will choose a strategy that maximizes their probabilities of winning. Assuming rational behavior and equal distribution, she would likely pick 50, which is the median number. This central strategy minimizes risk and maximizes the probability of being close to the randomly chosen number.

Choose Your Best Response

Knowing the opponent will likely choose 50, the best strategy is to choose numbers close to her choice. Therefore, picking either 49 or 51 maximizes the chances of winning, as it ensures proximity regardless of the random number's selection.

Analyze Closeness of the Choices

The numbers are close because both participants are aiming for a central, optimal choice that balances risk. Choosing numbers adjacent to the median minimizes the potential distance from the random number.

Relate to Business Strategy

This scenario mirrors business decisions in geographical location selection. Just as stores like McDonald's and Burger King locate near each other, competing businesses position themselves strategically to capture maximum market share while remaining close to competitors also targeting the median market 'customer.'

Key concepts

Strategic Decision Making

Strategic decision-making is about making choices that influence outcomes in a competitive environment. Imagine you are part of a game where both you and another person are trying to guess a number between 1 and 100 that's closest to a secretly chosen number. In this setup, your ability to make strategic decisions is tested.
To win, you must think several steps ahead and make educated predictions about your opponent's choice. This involves analyzing what they find most advantageous and selecting a number that positions you optimally in relation to their choice. The aim is to maximize your winning probability by carefully evaluating all possible outcomes. Similarly, in business, companies make strategic decisions by predicting competitor actions and adjusting their strategies accordingly.
Strategic decision-making requires:

• Anticipating how others will act
• Understanding the impact of your choices
• Finding the balance between risk and reward

Nash Equilibrium

The Nash Equilibrium concept in game theory is a situation where no player can benefit by changing their strategy while the other players' strategies remain unchanged. In our game of numbers, suppose your opponent chooses 50 as a strategy to maximize closeness to the random number. If you knew this, you'd choose either 49 or 51 to stay competitive.
Once both players have made these choices, neither can improve their outcome by picking a different number, given the other player's choice. This situation, where both sides have chosen optimal responses given the actions of the other, represents a Nash Equilibrium. You and your opponent have arrived at a stable strategy, neither willing to move away because it offers a maximum advantage.
Nash Equilibrium characteristics:

• Each player has chosen a best response to the other players' strategies
• No player gains by unilaterally changing their own decision
• Stable outcomes despite competitive settings

Business Strategy

In the realm of business strategy, understanding how to position your company relative to competitors is crucial. Drawing parallels to our example, businesses often face decisions similar to picking a number in a competitive game. Companies like McDonald's and Burger King often open stores close to one another. At first glance, this might seem counterintuitive.
However, similar strategies are designed to maximize exposure to a shared customer base and capture additional market share by being the most convenient choice. Businesses strategically decide on location, pricing, and product offerings, always taking into account the likely moves of their competitors.
Key aspects of business strategy derived from game theory include:

• Aligning positions relative to competitors
• Forecasting market trends and competitor behavior
• Optimizing operations for shared customer bases

Market Competition

Market competition is a significant aspect of economic environments where businesses strive to outdo each other while appealing to shared pools of customers. In the analogy of our numerical game, competitors are prompted to choose numbers strategically close to one another.
Businesses employ similar tactics by situating themselves in close geographical proximity to rivals. This move might appear as intensifying competition but in reality, it frequently leads to enhanced market attractiveness and benefits customers through increased choice.
Market competition involves:

• Companies sharing spaces to increase visibility
• Engaging in pricing wars and promotional strategies
• Bolstering customer loyalty through improved services and offers