Question 1178007: Suppose that you are attempting to buy a house, and you are bargaining with the current owner over the sale price. The house is of value $200,000 to you and $150,000 to the current owner; so, if the price is between $150,000 and $200,000, then you would both be better off with the sale. Assume that bargaining takes place with alternating offers and that each stage of bargaining (an offer and a response) takes a full day to complete. If an agreement is not reached after seven days of bargaining, then the opportunity for the sale disappears (you will have no house and the current owner has to keep the house forever). Suppose that you discount the future according to the discount factor 0.4 per day and the current owner discounts the future at a discount factor of 0.8 per day. The real estate agent has allowed you to decide whether you will make the first offer. Should you make the first offer or let the current owner make the first offer? Why? Suppose that you have to make the first offer, what will be your offer that is going to be acceptable to the current owner?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze this bargaining scenario step-by-step.
**Understanding the Problem**
* **Your Value:** $200,000
* **Owner's Value:** $150,000
* **Discount Factor (You):** 0.4 per day
* **Discount Factor (Owner):** 0.8 per day
* **Bargaining Period:** 7 days (14 rounds of offers and responses)
* **Alternating Offers:** You and the owner alternate making offers.
**Backward Induction**
We'll use backward induction to determine the outcome.
**Day 7 (Last Day):**
* If you make the last offer, you can offer slightly above $150,000, and the owner will accept (since it's better than nothing).
* If the owner makes the last offer, they can demand slightly below $200,000, and you will accept (since it's better than nothing).
**Day 6:**
* If you make the offer on Day 6, knowing the owner can get slightly above $150,000 on Day 7, you need to offer them a value that, when discounted, is equal to or greater than that.
* Let $P_7$ be the price the owner would accept on Day 7 (slightly above $150,000). To make it simple, let's assume $P_7 = $150,000.
* On Day 6, you need to offer $P_6$ such that 0.8 * $P_6$ ≥ $150,000.
* $P_6$ ≥ $150,000 / 0.8 = $187,500.
* If the owner makes the offer on Day 6, knowing you can get slightly below $200,000 on Day 7, they need to offer you a value that, when discounted, is equal to or greater than that.
* Let $P_7$ be the price you would accept on Day 7 (slightly below $200,000). Let's assume $P_7 = $200,000.
* On Day 6, they need to offer $P_6$ such that 0.4 * $P_6$ ≥ $200,000.
* $P_6$ ≥ $200,000 / 0.4 = $500,000 (This is impossible, meaning you will not agree to any offer on day 6)
**Generalizing the Pattern**
* **Owner's Discounting:** The owner's offers will be discounted by 0.8 each day.
* **Your Discounting:** Your offers will be discounted by 0.4 each day.
**Working Backwards**
* **Day 7 (Last Offer):**
* If you offer, you offer slightly above $150,000 (say, $150,001).
* If the owner offers, they offer slightly below $200,000 (say, $199,999).
* **Day 6:**
* If you offer, you offer $150,000 / 0.8 = $187,500.
* If the owner offers, you would need to accept $200,000 / 0.4 = $500,000. This is impossible, so you don't accept any offer.
* **Day 5:**
* If you offer, you offer $187,500 / 0.4 = $468,750 (impossible, no offer will be accepted)
* If the owner offers, they offer $187,500 * 0.8 = $150,000
* **Day 4:**
* If you offer, you offer $150,000 / 0.4 = $375,000 (impossible)
* If the owner offers, you need to accept $150,000 * 0.8 = $120,000
* **Day 3:**
* If you offer, you offer $120,000/0.8 = $150,000
* If the owner offers, you need to accept $375,000 * 0.4 = $150,000
* **Day 2:**
* If you offer, you offer $150,000 / 0.4 = $375,000
* If the owner offers, they need to accept $150,000 * 0.8 = $120,000
* **Day 1:**
* If you offer, you offer $120,000/0.8 = $150,000
* If the owner offers, you need to accept $375,000 * 0.4 = $150,000
**Conclusion**
* **Who Should Make the First Offer?**
* It doesn't matter who makes the first offer. The outcome will be the same. The offer will eventually settle on $150,000.
* **Your First Offer (If You Make It):**
* You should offer $150,000. This is the minimum price the owner will accept, and it's within your value range.
**Explanation**
The key is that your discount factor (0.4) is significantly lower than the owner's (0.8). This means you discount future offers much more heavily. As a result, you are willing to accept lower offers sooner. The bargaining will converge to the owner's minimum acceptable price ($150,000).
Final Answer: You should make the first offer or let the current owner make the first offer, because the outcome will be the same. If you have to make the first offer, your offer will be $150,000.
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