Finite Mathematics Quiz: Payoff Tables
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Payoff TablesQuestion 1 of 9

A logistics company evaluates three shipping strategies using a payoff table where entries represent cost savings in thousands of dollars. Strategy X dominates Strategy Y, and the minimax regret criterion selects Strategy Z. If Strategy X has the payoffs 40, 60, 80 and Strategy Z has payoffs 35, 70, 75, which statement about Strategy Y is necessarily true?

Strategy Y has at least one payoff greater than the corresponding payoff of Strategy X
Strategy Y has all payoffs less than the corresponding payoffs of Strategy X
Strategy Y's maximum regret exceeds Strategy Z's maximum regret by at least 5
Strategy Y achieves the maximum payoff in at least one state of nature
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Finite Mathematics Quiz

Finite Mathematics Quiz: Payoff Tables

Practice Payoff Tables in Finite Mathematics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Payoff Tables, giving you a quick way to practice the rules, question types, and explanations that matter most for Finite Mathematics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A logistics company evaluates three shipping strategies using a payoff table where entries represent cost savings in thousands of dollars. Strategy X dominates Strategy Y, and the minimax regret criterion selects Strategy Z. If Strategy X has the payoffs 40, 60, 80 and Strategy Z has payoffs 35, 70, 75, which statement about Strategy Y is necessarily true?

  1. Strategy Y has at least one payoff greater than the corresponding payoff of Strategy X
  2. Strategy Y has all payoffs less than the corresponding payoffs of Strategy X (correct answer)
  3. Strategy Y's maximum regret exceeds Strategy Z's maximum regret by at least 5
  4. Strategy Y achieves the maximum payoff in at least one state of nature
Explanation: If Strategy X dominates Strategy Y, then Strategy X has payoffs that are greater than or equal to Strategy Y's payoffs in every state of nature, with at least one being strictly greater. This means all of Strategy Y's payoffs must be less than the corresponding payoffs of Strategy X (40, 60, 80).

Question 2

A farmer constructs a payoff table for crop selection based on weather conditions, with payoffs representing profit per acre in dollars. The regret table shows that Crop A has regrets of 0, 50, 100 for dry, normal, and wet conditions respectively. If Crop B's payoffs are 200, 400, 300 for the same conditions, what is Crop A's payoff under normal weather conditions?

  1. 350 (correct answer)
  2. 400
  3. 450
  4. 500
Explanation: Regret is calculated as (maximum payoff for that condition) minus (strategy's payoff for that condition). Crop A has 0 regret under dry conditions, meaning it achieves the maximum payoff for dry conditions. Crop A has 50 regret under normal conditions, and Crop B pays 400 under normal conditions. Since Crop A's regret is 50, the maximum payoff under normal conditions must be at least 400. If the maximum is 400 (from Crop B), then Crop A's payoff = 400 - 50 = 350.

Question 3

A project manager is evaluating three software development approaches using a payoff table where values represent net present value in thousands of dollars. The equal likelihood criterion suggests Strategy 2, while the maximin criterion suggests Strategy 1. If Strategy 1 has payoffs of 20, 30, 40 and Strategy 3 has payoffs of 10, 35, 45, what can be concluded about Strategy 2's payoffs?

  1. Strategy 2's minimum payoff is less than 20, and its average payoff exceeds 30 (correct answer)
  2. Strategy 2's minimum payoff is greater than 20, and its average payoff is exactly 30
  3. Strategy 2's minimum payoff is less than 20, and its average payoff is less than 30
  4. Strategy 2's minimum payoff equals 20, and its average payoff exceeds 33.33
Explanation: Since maximin criterion chooses Strategy 1, Strategy 1 must have the highest minimum payoff (20). Therefore, Strategy 2's minimum payoff must be less than 20. Since equal likelihood criterion chooses Strategy 2, it must have the highest average payoff. Strategy 1's average is 30, and Strategy 3's average is 30, so Strategy 2's average must exceed 30.

Question 4

A concert promoter is planning an outdoor event. The promoter can rent a large tent for $5,000, a small tent for $2,000, or no tent for $0. The revenue from the event depends on the weather. If there is 'No Rain', the revenue will be $20,000 regardless of the tent choice. If it 'Rains', the revenue will be $10,000 with a large tent, $6,000 with a small tent, and only $1,000 with no tent.

For a decision analysis of this problem, what is the value in the opportunity loss (regret) table that corresponds to the action 'Rent Small Tent' and the state of nature 'Rain'?

  1. $2,000
  2. $4,000
  3. $5,000
  4. $1,000 (correct answer)
Explanation: When you encounter a decision analysis problem involving opportunity loss (regret), you're calculating how much worse each decision performs compared to the best possible decision for each scenario. To find the opportunity loss for "Rent Small Tent" when it "Rains," you first need to determine the net profit for each option in the rain scenario. Net profit equals revenue minus tent cost:
  • Large tent: $10,000 revenue - $5,000 cost = $5,000 profit
  • Small tent: $6,000 revenue - $2,000 cost = $4,000 profit
  • No tent: $1,000 revenue - $0 cost = $1,000 profit
The best decision when it rains is renting the large tent ($5,000 profit). The opportunity loss for renting a small tent is the difference between the best possible outcome and the small tent outcome: $5,000 - $4,000 = $1,000. Looking at the wrong answers: A) $2,000 is simply the cost of the small tent, not the opportunity loss. B) $4,000 is the actual profit from the small tent decision, not how much you lose by not choosing optimally. C) $5,000 is the profit from the optimal decision (large tent), but opportunity loss measures the gap between optimal and your chosen action. The correct answer is D) $1,000. Remember that opportunity loss always compares your chosen action against the best possible action for that specific scenario. It's not about costs or revenues directly—it's about the profit you're giving up by not making the optimal choice.

Question 5

A company manufactures a product at a cost of $30 per unit and sells it for $70 per unit. The company must decide whether to produce 100 units or 200 units. The market demand can be 100, 150, or 200 units. Any excess production can be sold for a salvage value of $10 per unit. If demand exceeds production, the company incurs a penalty of $5 per unit of unmet demand due to loss of goodwill.

In the payoff table for this situation, what is the payoff corresponding to producing 100 units when market demand is 200 units?

  1. $4,000
  2. $3,500 (correct answer)
  3. $1,000
  4. $3,000
Explanation:
  1. Calculate total revenue: All 100 units produced are sold. Revenue = 100 units * $70/unit = $7,000.
  2. Calculate total cost: Production cost = 100 units * $30/unit = $3,000.
  3. Calculate unmet demand: Demand (200) - Production (100) = 100 units.
  4. Calculate penalty cost: 100 units unmet demand * $5/unit = $500.
  5. Calculate final payoff (profit): Revenue - Production Cost - Penalty Cost = $7,000 - $3,000 - $500 = $3,500.

Question 6

A tech startup has developed a new app. The initial development cost was $250,000. The company must now decide whether to launch the app ('Launch') or sell the technology to a larger company ('Sell'). The 'Sell' option guarantees a payment of $300,000. The outcome of the 'Launch' decision depends on market adoption, which can be 'Low' or 'High'. If adoption is 'Low', the app will generate revenues of $100,000. If adoption is 'High', it will generate revenues of $800,000. The payoff is defined as the final financial position, considering all revenues and the initial cost.

In the payoff table for this decision, what is the payoff for the 'Launch' alternative if market adoption is 'Low'?

  1. -$250,000
  2. $100,000
  3. -$150,000 (correct answer)
  4. $50,000
Explanation: When analyzing payoff tables in decision theory, you need to understand that payoff represents the net financial outcome after considering all costs and revenues. This means you must subtract initial investments from any revenues generated. For the 'Launch' decision with 'Low' market adoption, you need to calculate the net payoff by taking the revenue generated (100,000)andsubtractingtheinitialdevelopmentcost(100,000) and subtracting the initial development cost (250,000). This gives you: $100,000 - 250,000=−250,000 = -150,000. The negative value indicates a net loss. Let's examine why the other options are incorrect. Choice A (-$250,000) represents only the initial cost without accounting for any revenue generated from the low adoption scenario. This ignores the 100,000inrevenuetheappwouldstillgenerate.ChoiceB(100,000 in revenue the app would still generate. Choice B (100,000) is simply the gross revenue figure and fails to account for the initial development cost that must be recovered. Choice D ($50,000) appears to be a calculation error, possibly mixing up numbers or using incorrect arithmetic. The key insight here is that payoff tables always show net outcomes, not gross revenues. Even when market adoption is low, the company still generates some revenue, but it's not enough to cover the initial investment, resulting in an overall loss. Remember: in payoff calculations, always subtract all relevant costs from revenues to get the true financial position. Don't confuse gross revenue with net payoff – the initial investment must always be factored into your final calculation.

Question 7

A farmer is deciding which of three crops to plant: corn, soybeans, or wheat. The profit per acre depends on the weather, which can be 'Dry', 'Moderate', or 'Wet'. The farmer estimates the following profits per acre:\n- Corn: $100 if Dry, $200 if Moderate, $50 if Wet.\n- Soybeans: $120 if Dry, $150 if Moderate, $130 if Wet.\n- Wheat: $150 if Dry, $140 if Moderate, $80 if Wet.\nThe farmer decides to construct a payoff table to analyze the decision.

In the payoff table representing this problem, which value correctly represents the payoff for a specific crop under a specific weather condition?

  1. The payoff for planting soybeans in Wet weather is $150.
  2. The payoff for planting corn in Dry weather is $100. (correct answer)
  3. The payoff for planting wheat in Moderate weather is $200.
  4. The payoff for planting corn in Wet weather is $120.
Explanation: The question requires correctly identifying an entry in the payoff table based on the text. A payoff table for this problem would have the crops (Corn, Soybeans, Wheat) as the alternatives (rows) and the weather conditions (Dry, Moderate, Wet) as the states of nature (columns). Reading the provided data:\n- The profit for Corn if Dry is $100. This matches choice B.\n- The profit for Soybeans in Wet weather is $130 (not $150). So A is incorrect.\n- The profit for Wheat in Moderate weather is $140 (not $200). So C is incorrect.\n- The profit for Corn in Wet weather is $50 (not $120). So D is incorrect.

Question 8

A specialty bakery plans to sell a seasonal cake. The owner must decide whether to bake a small batch of 50 cakes or a large batch of 80 cakes. Each cake costs $15 to make and sells for $40. Any cakes not sold by the end of the season are sold to a discount outlet for a salvage value of $5 each. The demand for the cake can be either low (50 cakes) or high (80 cakes).

In the payoff table for this decision problem, what is the profit (payoff) if the bakery produces the large batch and the demand is low?

  1. $850 (correct answer)
  2. $1000
  3. $2000
  4. $800
Explanation: To calculate the payoff for producing the large batch (80 cakes) with low demand (50 cakes):
  1. Calculate the production cost: 80 cakes * $15/cake = $1200.
  2. Calculate the revenue from sold cakes: 50 cakes * $40/cake = $2000.
  3. Calculate the number of unsold cakes: 80 cakes produced - 50 cakes sold = 30 cakes.
  4. Calculate the revenue from salvaged cakes: 30 cakes * $5/cake = $150.
  5. Calculate total revenue: $2000 + $150 = $2150.
  6. Calculate the profit (payoff): Total Revenue - Total Cost = $2150 - $1200 = $850.

Question 9

A retailer is constructing a payoff table for ordering seasonal merchandise. She has three ordering options (Small, Medium, Large) and three possible demand scenarios. The payoffs are given as profits in hundreds of dollars. If the Medium order yields 15 under low demand, 25 under medium demand, and 30 under high demand, and the retailer wants to identify the strategy that maximizes the minimum possible payoff, what additional information is most critical for completing this analysis?

  1. The probability distribution of the demand scenarios occurring in future periods
  2. The payoffs for Small and Large order strategies under all demand scenarios (correct answer)
  3. The opportunity costs associated with stockout situations for each strategy
  4. The historical frequency data for similar seasonal merchandise categories
Explanation: To apply the maximin criterion (maximize the minimum payoff), we need the complete payoff table showing all strategies under all scenarios. The maximin criterion only requires the payoff values, not probabilities (A), opportunity costs beyond those reflected in payoffs (C), or historical data (D).