1. **Problem Statement:** A Coca-Cola distributor buys bottles for 6 and sells them for 10 each. Leftover bottles are worthless. Daily sales range from 10 to 12. Prepare the payoff and regret tables. 2. **Formula and Rules:** - Payoff = (Selling price × number sold) - (Cost price × number bought) - Leftover bottles have zero salvage value. - Sales can be 10, 11, or 12 bottles. - The distributor decides how many bottles to buy (10, 11, or 12). 3. **Constructing the Payoff Table:** Let $x$ = bottles bought, $s$ = bottles sold. Payoff $= 10 \times \min(x,s) - 6x$ Calculate for each $x$ and $s$: - Buy 10: - Sell 10: $10 \times 10 - 6 \times 10 = 100 - 60 = 40$ - Sell 11: $10 \times 10 - 60 = 40$ - Sell 12: $10 \times 10 - 60 = 40$ - Buy 11: - Sell 10: $10 \times 10 - 6 \times 11 = 100 - 66 = 34$ - Sell 11: $10 \times 11 - 66 = 110 - 66 = 44$ - Sell 12: $10 \times 11 - 66 = 44$ - Buy 12: - Sell 10: $10 \times 10 - 6 \times 12 = 100 - 72 = 28$ - Sell 11: $10 \times 11 - 72 = 110 - 72 = 38$ - Sell 12: $10 \times 12 - 72 = 120 - 72 = 48$ Payoff Table: | Buy/Sell | 10 | 11 | 12 | |----------|----|----|----| | 10 | 40 | 40 | 40 | | 11 | 34 | 44 | 44 | | 12 | 28 | 38 | 48 | 4. **Constructing the Regret Table:** Regret = Maximum payoff for each sales level - payoff for each buy decision. - For sales = 10, max payoff = 40 - For sales = 11, max payoff = 44 - For sales = 12, max payoff = 48 Calculate regrets: - Buy 10: - Sell 10: $40 - 40 = 0$ - Sell 11: $44 - 40 = 4$ - Sell 12: $48 - 40 = 8$ - Buy 11: - Sell 10: $40 - 34 = 6$ - Sell 11: $44 - 44 = 0$ - Sell 12: $48 - 44 = 4$ - Buy 12: - Sell 10: $40 - 28 = 12$ - Sell 11: $44 - 38 = 6$ - Sell 12: $48 - 48 = 0$ Regret Table: | Buy/Sell | 10 | 11 | 12 | |----------|----|----|----| | 10 | 0 | 4 | 8 | | 11 | 6 | 0 | 4 | | 12 | 12 | 6 | 0 | **Final Answer:** Payoff and regret tables are as above, helping the distributor decide the optimal number of bottles to buy based on sales uncertainty.